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# Copyright 2002 Gary Strangman. All rights reserved
# Copyright 2002-2016 The SciPy Developers
#
# The original code from Gary Strangman was heavily adapted for
# use in SciPy by Travis Oliphant. The original code came with the
# following disclaimer:
#
# This software is provided "as-is". There are no expressed or implied
# warranties of any kind, including, but not limited to, the warranties
# of merchantability and fitness for a given application. In no event
# shall Gary Strangman be liable for any direct, indirect, incidental,
# special, exemplary or consequential damages (including, but not limited
# to, loss of use, data or profits, or business interruption) however
# caused and on any theory of liability, whether in contract, strict
# liability or tort (including negligence or otherwise) arising in any way
# out of the use of this software, even if advised of the possibility of
# such damage.
"""
A collection of basic statistical functions for Python. The function
names appear below.
Some scalar functions defined here are also available in the scipy.special
package where they work on arbitrary sized arrays.
Disclaimers: The function list is obviously incomplete and, worse, the
functions are not optimized. All functions have been tested (some more
so than others), but they are far from bulletproof. Thus, as with any
free software, no warranty or guarantee is expressed or implied. :-) A
few extra functions that don't appear in the list below can be found by
interested treasure-hunters. These functions don't necessarily have
both list and array versions but were deemed useful.
Central Tendency
----------------
.. autosummary::
:toctree: generated/
gmean
hmean
mode
Moments
-------
.. autosummary::
:toctree: generated/
moment
variation
skew
kurtosis
normaltest
Altered Versions
----------------
.. autosummary::
:toctree: generated/
tmean
tvar
tstd
tsem
describe
Frequency Stats
---------------
.. autosummary::
:toctree: generated/
itemfreq
scoreatpercentile
percentileofscore
cumfreq
relfreq
Variability
-----------
.. autosummary::
:toctree: generated/
obrientransform
sem
zmap
zscore
iqr
Trimming Functions
------------------
.. autosummary::
:toctree: generated/
trimboth
trim1
Correlation Functions
---------------------
.. autosummary::
:toctree: generated/
pearsonr
fisher_exact
spearmanr
pointbiserialr
kendalltau
weightedtau
linregress
theilslopes
Inferential Stats
-----------------
.. autosummary::
:toctree: generated/
ttest_1samp
ttest_ind
ttest_ind_from_stats
ttest_rel
chisquare
power_divergence
ks_2samp
mannwhitneyu
ranksums
wilcoxon
kruskal
friedmanchisquare
combine_pvalues
Statistical Distances
---------------------
.. autosummary::
:toctree: generated/
wasserstein_distance
energy_distance
ANOVA Functions
---------------
.. autosummary::
:toctree: generated/
f_oneway
Support Functions
-----------------
.. autosummary::
:toctree: generated/
rankdata
References
----------
.. [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
"""
from __future__ import division, print_function, absolute_import
import warnings
import math
from collections import namedtuple
import numpy as np
from numpy import array, asarray, ma, zeros
from scipy._lib.six import callable, string_types
from scipy._lib._version import NumpyVersion
import scipy.special as special
import scipy.linalg as linalg
from . import distributions
from . import mstats_basic
from ._distn_infrastructure import _lazywhere
from ._stats_mstats_common import _find_repeats, linregress, theilslopes
from ._stats import _kendall_dis, _toint64, _weightedrankedtau
__all__ = ['find_repeats', 'gmean', 'hmean', 'mode', 'tmean', 'tvar',
'tmin', 'tmax', 'tstd', 'tsem', 'moment', 'variation',
'skew', 'kurtosis', 'describe', 'skewtest', 'kurtosistest',
'normaltest', 'jarque_bera', 'itemfreq',
'scoreatpercentile', 'percentileofscore',
'cumfreq', 'relfreq', 'obrientransform',
'sem', 'zmap', 'zscore', 'iqr',
'sigmaclip', 'trimboth', 'trim1', 'trim_mean', 'f_oneway',
'pearsonr', 'fisher_exact', 'spearmanr', 'pointbiserialr',
'kendalltau', 'weightedtau',
'linregress', 'theilslopes', 'ttest_1samp',
'ttest_ind', 'ttest_ind_from_stats', 'ttest_rel', 'kstest',
'chisquare', 'power_divergence', 'ks_2samp', 'mannwhitneyu',
'tiecorrect', 'ranksums', 'kruskal', 'friedmanchisquare',
'rankdata',
'combine_pvalues', 'wasserstein_distance', 'energy_distance']
def _chk_asarray(a, axis):
if axis is None:
a = np.ravel(a)
outaxis = 0
else:
a = np.asarray(a)
outaxis = axis
if a.ndim == 0:
a = np.atleast_1d(a)
return a, outaxis
def _chk2_asarray(a, b, axis):
if axis is None:
a = np.ravel(a)
b = np.ravel(b)
outaxis = 0
else:
a = np.asarray(a)
b = np.asarray(b)
outaxis = axis
if a.ndim == 0:
a = np.atleast_1d(a)
if b.ndim == 0:
b = np.atleast_1d(b)
return a, b, outaxis
def _contains_nan(a, nan_policy='propagate'):
policies = ['propagate', 'raise', 'omit']
if nan_policy not in policies:
raise ValueError("nan_policy must be one of {%s}" %
', '.join("'%s'" % s for s in policies))
try:
# Calling np.sum to avoid creating a huge array into memory
# e.g. np.isnan(a).any()
with np.errstate(invalid='ignore'):
contains_nan = np.isnan(np.sum(a))
except TypeError:
# If the check cannot be properly performed we fallback to omitting
# nan values and raising a warning. This can happen when attempting to
# sum things that are not numbers (e.g. as in the function `mode`).
contains_nan = False
nan_policy = 'omit'
warnings.warn("The input array could not be properly checked for nan "
"values. nan values will be ignored.", RuntimeWarning)
if contains_nan and nan_policy == 'raise':
raise ValueError("The input contains nan values")
return (contains_nan, nan_policy)
def gmean(a, axis=0, dtype=None):
"""
Compute the geometric mean along the specified axis.
Return the geometric average of the array elements.
That is: n-th root of (x1 * x2 * ... * xn)
Parameters
----------
a : array_like
Input array or object that can be converted to an array.
axis : int or None, optional
Axis along which the geometric mean is computed. Default is 0.
If None, compute over the whole array `a`.
dtype : dtype, optional
Type of the returned array and of the accumulator in which the
elements are summed. If dtype is not specified, it defaults to the
dtype of a, unless a has an integer dtype with a precision less than
that of the default platform integer. In that case, the default
platform integer is used.
Returns
-------
gmean : ndarray
see dtype parameter above
See Also
--------
numpy.mean : Arithmetic average
numpy.average : Weighted average
hmean : Harmonic mean
Notes
-----
The geometric average is computed over a single dimension of the input
array, axis=0 by default, or all values in the array if axis=None.
float64 intermediate and return values are used for integer inputs.
Use masked arrays to ignore any non-finite values in the input or that
arise in the calculations such as Not a Number and infinity because masked
arrays automatically mask any non-finite values.
Examples
--------
>>> from scipy.stats import gmean
>>> gmean([1, 4])
2.0
>>> gmean([1, 2, 3, 4, 5, 6, 7])
3.3800151591412964
"""
if not isinstance(a, np.ndarray):
# if not an ndarray object attempt to convert it
log_a = np.log(np.array(a, dtype=dtype))
elif dtype:
# Must change the default dtype allowing array type
if isinstance(a, np.ma.MaskedArray):
log_a = np.log(np.ma.asarray(a, dtype=dtype))
else:
log_a = np.log(np.asarray(a, dtype=dtype))
else:
log_a = np.log(a)
return np.exp(log_a.mean(axis=axis))
def hmean(a, axis=0, dtype=None):
"""
Calculate the harmonic mean along the specified axis.
That is: n / (1/x1 + 1/x2 + ... + 1/xn)
Parameters
----------
a : array_like
Input array, masked array or object that can be converted to an array.
axis : int or None, optional
Axis along which the harmonic mean is computed. Default is 0.
If None, compute over the whole array `a`.
dtype : dtype, optional
Type of the returned array and of the accumulator in which the
elements are summed. If `dtype` is not specified, it defaults to the
dtype of `a`, unless `a` has an integer `dtype` with a precision less
than that of the default platform integer. In that case, the default
platform integer is used.
Returns
-------
hmean : ndarray
see `dtype` parameter above
See Also
--------
numpy.mean : Arithmetic average
numpy.average : Weighted average
gmean : Geometric mean
Notes
-----
The harmonic mean is computed over a single dimension of the input
array, axis=0 by default, or all values in the array if axis=None.
float64 intermediate and return values are used for integer inputs.
Use masked arrays to ignore any non-finite values in the input or that
arise in the calculations such as Not a Number and infinity.
Examples
--------
>>> from scipy.stats import hmean
>>> hmean([1, 4])
1.6000000000000001
>>> hmean([1, 2, 3, 4, 5, 6, 7])
2.6997245179063363
"""
if not isinstance(a, np.ndarray):
a = np.array(a, dtype=dtype)
if np.all(a > 0):
# Harmonic mean only defined if greater than zero
if isinstance(a, np.ma.MaskedArray):
size = a.count(axis)
else:
if axis is None:
a = a.ravel()
size = a.shape[0]
else:
size = a.shape[axis]
return size / np.sum(1.0 / a, axis=axis, dtype=dtype)
else:
raise ValueError("Harmonic mean only defined if all elements greater "
"than zero")
ModeResult = namedtuple('ModeResult', ('mode', 'count'))
def mode(a, axis=0, nan_policy='propagate'):
"""
Return an array of the modal (most common) value in the passed array.
If there is more than one such value, only the smallest is returned.
The bin-count for the modal bins is also returned.
Parameters
----------
a : array_like
n-dimensional array of which to find mode(s).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over
the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
mode : ndarray
Array of modal values.
count : ndarray
Array of counts for each mode.
Examples
--------
>>> a = np.array([[6, 8, 3, 0],
... [3, 2, 1, 7],
... [8, 1, 8, 4],
... [5, 3, 0, 5],
... [4, 7, 5, 9]])
>>> from scipy import stats
>>> stats.mode(a)
(array([[3, 1, 0, 0]]), array([[1, 1, 1, 1]]))
To get mode of whole array, specify ``axis=None``:
>>> stats.mode(a, axis=None)
(array([3]), array([3]))
"""
a, axis = _chk_asarray(a, axis)
if a.size == 0:
return ModeResult(np.array([]), np.array([]))
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.mode(a, axis)
scores = np.unique(np.ravel(a)) # get ALL unique values
testshape = list(a.shape)
testshape[axis] = 1
oldmostfreq = np.zeros(testshape, dtype=a.dtype)
oldcounts = np.zeros(testshape, dtype=int)
for score in scores:
template = (a == score)
counts = np.expand_dims(np.sum(template, axis), axis)
mostfrequent = np.where(counts > oldcounts, score, oldmostfreq)
oldcounts = np.maximum(counts, oldcounts)
oldmostfreq = mostfrequent
return ModeResult(mostfrequent, oldcounts)
def _mask_to_limits(a, limits, inclusive):
"""Mask an array for values outside of given limits.
This is primarily a utility function.
Parameters
----------
a : array
limits : (float or None, float or None)
A tuple consisting of the (lower limit, upper limit). Values in the
input array less than the lower limit or greater than the upper limit
will be masked out. None implies no limit.
inclusive : (bool, bool)
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to lower or upper are allowed.
Returns
-------
A MaskedArray.
Raises
------
A ValueError if there are no values within the given limits.
"""
lower_limit, upper_limit = limits
lower_include, upper_include = inclusive
am = ma.MaskedArray(a)
if lower_limit is not None:
if lower_include:
am = ma.masked_less(am, lower_limit)
else:
am = ma.masked_less_equal(am, lower_limit)
if upper_limit is not None:
if upper_include:
am = ma.masked_greater(am, upper_limit)
else:
am = ma.masked_greater_equal(am, upper_limit)
if am.count() == 0:
raise ValueError("No array values within given limits")
return am
def tmean(a, limits=None, inclusive=(True, True), axis=None):
"""
Compute the trimmed mean.
This function finds the arithmetic mean of given values, ignoring values
outside the given `limits`.
Parameters
----------
a : array_like
Array of values.
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None (default), then all
values are used. Either of the limit values in the tuple can also be
None representing a half-open interval.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to compute test. Default is None.
Returns
-------
tmean : float
See also
--------
trim_mean : returns mean after trimming a proportion from both tails.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tmean(x)
9.5
>>> stats.tmean(x, (3,17))
10.0
"""
a = asarray(a)
if limits is None:
return np.mean(a, None)
am = _mask_to_limits(a.ravel(), limits, inclusive)
return am.mean(axis=axis)
def tvar(a, limits=None, inclusive=(True, True), axis=0, ddof=1):
"""
Compute the trimmed variance.
This function computes the sample variance of an array of values,
while ignoring values which are outside of given `limits`.
Parameters
----------
a : array_like
Array of values.
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
ddof : int, optional
Delta degrees of freedom. Default is 1.
Returns
-------
tvar : float
Trimmed variance.
Notes
-----
`tvar` computes the unbiased sample variance, i.e. it uses a correction
factor ``n / (n - 1)``.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tvar(x)
35.0
>>> stats.tvar(x, (3,17))
20.0
"""
a = asarray(a)
a = a.astype(float).ravel()
if limits is None:
n = len(a)
return a.var() * n / (n - 1.)
am = _mask_to_limits(a, limits, inclusive)
return np.ma.var(am, ddof=ddof, axis=axis)
def tmin(a, lowerlimit=None, axis=0, inclusive=True, nan_policy='propagate'):
"""
Compute the trimmed minimum.
This function finds the miminum value of an array `a` along the
specified axis, but only considering values greater than a specified
lower limit.
Parameters
----------
a : array_like
array of values
lowerlimit : None or float, optional
Values in the input array less than the given limit will be ignored.
When lowerlimit is None, then all values are used. The default value
is None.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the lower limit
are included. The default value is True.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
tmin : float, int or ndarray
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tmin(x)
0
>>> stats.tmin(x, 13)
13
>>> stats.tmin(x, 13, inclusive=False)
14
"""
a, axis = _chk_asarray(a, axis)
am = _mask_to_limits(a, (lowerlimit, None), (inclusive, False))
contains_nan, nan_policy = _contains_nan(am, nan_policy)
if contains_nan and nan_policy == 'omit':
am = ma.masked_invalid(am)
res = ma.minimum.reduce(am, axis).data
if res.ndim == 0:
return res[()]
return res
def tmax(a, upperlimit=None, axis=0, inclusive=True, nan_policy='propagate'):
"""
Compute the trimmed maximum.
This function computes the maximum value of an array along a given axis,
while ignoring values larger than a specified upper limit.
Parameters
----------
a : array_like
array of values
upperlimit : None or float, optional
Values in the input array greater than the given limit will be ignored.
When upperlimit is None, then all values are used. The default value
is None.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the upper limit
are included. The default value is True.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
tmax : float, int or ndarray
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tmax(x)
19
>>> stats.tmax(x, 13)
13
>>> stats.tmax(x, 13, inclusive=False)
12
"""
a, axis = _chk_asarray(a, axis)
am = _mask_to_limits(a, (None, upperlimit), (False, inclusive))
contains_nan, nan_policy = _contains_nan(am, nan_policy)
if contains_nan and nan_policy == 'omit':
am = ma.masked_invalid(am)
res = ma.maximum.reduce(am, axis).data
if res.ndim == 0:
return res[()]
return res
def tstd(a, limits=None, inclusive=(True, True), axis=0, ddof=1):
"""
Compute the trimmed sample standard deviation.
This function finds the sample standard deviation of given values,
ignoring values outside the given `limits`.
Parameters
----------
a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
ddof : int, optional
Delta degrees of freedom. Default is 1.
Returns
-------
tstd : float
Notes
-----
`tstd` computes the unbiased sample standard deviation, i.e. it uses a
correction factor ``n / (n - 1)``.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tstd(x)
5.9160797830996161
>>> stats.tstd(x, (3,17))
4.4721359549995796
"""
return np.sqrt(tvar(a, limits, inclusive, axis, ddof))
def tsem(a, limits=None, inclusive=(True, True), axis=0, ddof=1):
"""
Compute the trimmed standard error of the mean.
This function finds the standard error of the mean for given
values, ignoring values outside the given `limits`.
Parameters
----------
a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the
whole array `a`.
ddof : int, optional
Delta degrees of freedom. Default is 1.
Returns
-------
tsem : float
Notes
-----
`tsem` uses unbiased sample standard deviation, i.e. it uses a
correction factor ``n / (n - 1)``.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tsem(x)
1.3228756555322954
>>> stats.tsem(x, (3,17))
1.1547005383792515
"""
a = np.asarray(a).ravel()
if limits is None:
return a.std(ddof=ddof) / np.sqrt(a.size)
am = _mask_to_limits(a, limits, inclusive)
sd = np.sqrt(np.ma.var(am, ddof=ddof, axis=axis))
return sd / np.sqrt(am.count())
#####################################
# MOMENTS #
#####################################
def moment(a, moment=1, axis=0, nan_policy='propagate'):
r"""
Calculate the nth moment about the mean for a sample.
A moment is a specific quantitative measure of the shape of a set of
points. It is often used to calculate coefficients of skewness and kurtosis
due to its close relationship with them.
Parameters
----------
a : array_like
data
moment : int or array_like of ints, optional
order of central moment that is returned. Default is 1.
axis : int or None, optional
Axis along which the central moment is computed. Default is 0.
If None, compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
n-th central moment : ndarray or float
The appropriate moment along the given axis or over all values if axis
is None. The denominator for the moment calculation is the number of
observations, no degrees of freedom correction is done.
See also
--------
kurtosis, skew, describe
Notes
-----
The k-th central moment of a data sample is:
.. math::
m_k = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^k
Where n is the number of samples and x-bar is the mean. This function uses
exponentiation by squares [1]_ for efficiency.
References
----------
.. [1] http://eli.thegreenplace.net/2009/03/21/efficient-integer-exponentiation-algorithms
Examples
--------
>>> from scipy.stats import moment
>>> moment([1, 2, 3, 4, 5], moment=1)
0.0
>>> moment([1, 2, 3, 4, 5], moment=2)
2.0
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.moment(a, moment, axis)
if a.size == 0:
# empty array, return nan(s) with shape matching `moment`
if np.isscalar(moment):
return np.nan
else:
return np.ones(np.asarray(moment).shape, dtype=np.float64) * np.nan
# for array_like moment input, return a value for each.
if not np.isscalar(moment):
mmnt = [_moment(a, i, axis) for i in moment]
return np.array(mmnt)
else:
return _moment(a, moment, axis)
def _moment(a, moment, axis):
if np.abs(moment - np.round(moment)) > 0:
raise ValueError("All moment parameters must be integers")
if moment == 0:
# When moment equals 0, the result is 1, by definition.
shape = list(a.shape)
del shape[axis]
if shape:
# return an actual array of the appropriate shape
return np.ones(shape, dtype=float)
else:
# the input was 1D, so return a scalar instead of a rank-0 array
return 1.0
elif moment == 1:
# By definition the first moment about the mean is 0.
shape = list(a.shape)
del shape[axis]
if shape:
# return an actual array of the appropriate shape
return np.zeros(shape, dtype=float)
else:
# the input was 1D, so return a scalar instead of a rank-0 array
return np.float64(0.0)
else:
# Exponentiation by squares: form exponent sequence
n_list = [moment]
current_n = moment
while current_n > 2:
if current_n % 2:
current_n = (current_n - 1) / 2
else:
current_n /= 2
n_list.append(current_n)
# Starting point for exponentiation by squares
a_zero_mean = a - np.expand_dims(np.mean(a, axis), axis)
if n_list[-1] == 1:
s = a_zero_mean.copy()
else:
s = a_zero_mean**2
# Perform multiplications
for n in n_list[-2::-1]:
s = s**2
if n % 2:
s *= a_zero_mean
return np.mean(s, axis)
def variation(a, axis=0, nan_policy='propagate'):
"""
Compute the coefficient of variation, the ratio of the biased standard
deviation to the mean.
Parameters
----------
a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate the coefficient of variation. Default
is 0. If None, compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
variation : ndarray
The calculated variation along the requested axis.
References
----------
.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
Examples
--------
>>> from scipy.stats import variation
>>> variation([1, 2, 3, 4, 5])
0.47140452079103173
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.variation(a, axis)
return a.std(axis) / a.mean(axis)
def skew(a, axis=0, bias=True, nan_policy='propagate'):
"""
Compute the skewness of a data set.
For normally distributed data, the skewness should be about 0. For
unimodal continuous distributions, a skewness value > 0 means that
there is more weight in the right tail of the distribution. The
function `skewtest` can be used to determine if the skewness value
is close enough to 0, statistically speaking.
Parameters
----------
a : ndarray
data
axis : int or None, optional
Axis along which skewness is calculated. Default is 0.
If None, compute over the whole array `a`.
bias : bool, optional
If False, then the calculations are corrected for statistical bias.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
skewness : ndarray
The skewness of values along an axis, returning 0 where all values are
equal.
References
----------
.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
Section 2.2.24.1
Examples
--------
>>> from scipy.stats import skew
>>> skew([1, 2, 3, 4, 5])
0.0
>>> skew([2, 8, 0, 4, 1, 9, 9, 0])
0.2650554122698573
"""
a, axis = _chk_asarray(a, axis)
n = a.shape[axis]
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.skew(a, axis, bias)
m2 = moment(a, 2, axis)
m3 = moment(a, 3, axis)
zero = (m2 == 0)
vals = _lazywhere(~zero, (m2, m3),
lambda m2, m3: m3 / m2**1.5,
0.)
if not bias:
can_correct = (n > 2) & (m2 > 0)
if can_correct.any():
m2 = np.extract(can_correct, m2)
m3 = np.extract(can_correct, m3)
nval = np.sqrt((n - 1.0) * n) / (n - 2.0) * m3 / m2**1.5
np.place(vals, can_correct, nval)
if vals.ndim == 0:
return vals.item()
return vals
def kurtosis(a, axis=0, fisher=True, bias=True, nan_policy='propagate'):
"""
Compute the kurtosis (Fisher or Pearson) of a dataset.
Kurtosis is the fourth central moment divided by the square of the
variance. If Fisher's definition is used, then 3.0 is subtracted from
the result to give 0.0 for a normal distribution.
If bias is False then the kurtosis is calculated using k statistics to
eliminate bias coming from biased moment estimators
Use `kurtosistest` to see if result is close enough to normal.
Parameters
----------
a : array
data for which the kurtosis is calculated
axis : int or None, optional
Axis along which the kurtosis is calculated. Default is 0.
If None, compute over the whole array `a`.
fisher : bool, optional
If True, Fisher's definition is used (normal ==> 0.0). If False,
Pearson's definition is used (normal ==> 3.0).
bias : bool, optional
If False, then the calculations are corrected for statistical bias.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
kurtosis : array
The kurtosis of values along an axis. If all values are equal,
return -3 for Fisher's definition and 0 for Pearson's definition.
References
----------
.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
Examples
--------
>>> from scipy.stats import kurtosis
>>> kurtosis([1, 2, 3, 4, 5])
-1.3
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.kurtosis(a, axis, fisher, bias)
n = a.shape[axis]
m2 = moment(a, 2, axis)
m4 = moment(a, 4, axis)
zero = (m2 == 0)
olderr = np.seterr(all='ignore')
try:
vals = np.where(zero, 0, m4 / m2**2.0)
finally:
np.seterr(**olderr)
if not bias:
can_correct = (n > 3) & (m2 > 0)
if can_correct.any():
m2 = np.extract(can_correct, m2)
m4 = np.extract(can_correct, m4)
nval = 1.0/(n-2)/(n-3) * ((n**2-1.0)*m4/m2**2.0 - 3*(n-1)**2.0)
np.place(vals, can_correct, nval + 3.0)
if vals.ndim == 0:
vals = vals.item() # array scalar
if fisher:
return vals - 3
else:
return vals
DescribeResult = namedtuple('DescribeResult',
('nobs', 'minmax', 'mean', 'variance', 'skewness',
'kurtosis'))
def describe(a, axis=0, ddof=1, bias=True, nan_policy='propagate'):
"""
Compute several descriptive statistics of the passed array.
Parameters
----------
a : array_like
Input data.
axis : int or None, optional
Axis along which statistics are calculated. Default is 0.
If None, compute over the whole array `a`.
ddof : int, optional
Delta degrees of freedom (only for variance). Default is 1.
bias : bool, optional
If False, then the skewness and kurtosis calculations are corrected for
statistical bias.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
nobs : int or ndarray of ints
Number of observations (length of data along `axis`).
When 'omit' is chosen as nan_policy, each column is counted separately.
minmax: tuple of ndarrays or floats
Minimum and maximum value of data array.
mean : ndarray or float
Arithmetic mean of data along axis.
variance : ndarray or float
Unbiased variance of the data along axis, denominator is number of
observations minus one.
skewness : ndarray or float
Skewness, based on moment calculations with denominator equal to
the number of observations, i.e. no degrees of freedom correction.
kurtosis : ndarray or float
Kurtosis (Fisher). The kurtosis is normalized so that it is
zero for the normal distribution. No degrees of freedom are used.
See Also
--------
skew, kurtosis
Examples
--------
>>> from scipy import stats
>>> a = np.arange(10)
>>> stats.describe(a)
DescribeResult(nobs=10, minmax=(0, 9), mean=4.5, variance=9.166666666666666,
skewness=0.0, kurtosis=-1.2242424242424244)
>>> b = [[1, 2], [3, 4]]
>>> stats.describe(b)
DescribeResult(nobs=2, minmax=(array([1, 2]), array([3, 4])),
mean=array([2., 3.]), variance=array([2., 2.]),
skewness=array([0., 0.]), kurtosis=array([-2., -2.]))
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.describe(a, axis, ddof, bias)
if a.size == 0:
raise ValueError("The input must not be empty.")
n = a.shape[axis]
mm = (np.min(a, axis=axis), np.max(a, axis=axis))
m = np.mean(a, axis=axis)
v = np.var(a, axis=axis, ddof=ddof)
sk = skew(a, axis, bias=bias)
kurt = kurtosis(a, axis, bias=bias)
return DescribeResult(n, mm, m, v, sk, kurt)
#####################################
# NORMALITY TESTS #
#####################################
SkewtestResult = namedtuple('SkewtestResult', ('statistic', 'pvalue'))
def skewtest(a, axis=0, nan_policy='propagate'):
"""
Test whether the skew is different from the normal distribution.
This function tests the null hypothesis that the skewness of
the population that the sample was drawn from is the same
as that of a corresponding normal distribution.
Parameters
----------
a : array
The data to be tested
axis : int or None, optional
Axis along which statistics are calculated. Default is 0.
If None, compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float
The computed z-score for this test.
pvalue : float
a 2-sided p-value for the hypothesis test
Notes
-----
The sample size must be at least 8.
References
----------
.. [1] R. B. D'Agostino, A. J. Belanger and R. B. D'Agostino Jr.,
"A suggestion for using powerful and informative tests of
normality", American Statistician 44, pp. 316-321, 1990.
Examples
--------
>>> from scipy.stats import skewtest
>>> skewtest([1, 2, 3, 4, 5, 6, 7, 8])
SkewtestResult(statistic=1.0108048609177787, pvalue=0.3121098361421897)
>>> skewtest([2, 8, 0, 4, 1, 9, 9, 0])
SkewtestResult(statistic=0.44626385374196975, pvalue=0.6554066631275459)
>>> skewtest([1, 2, 3, 4, 5, 6, 7, 8000])
SkewtestResult(statistic=3.571773510360407, pvalue=0.0003545719905823133)
>>> skewtest([100, 100, 100, 100, 100, 100, 100, 101])
SkewtestResult(statistic=3.5717766638478072, pvalue=0.000354567720281634)
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.skewtest(a, axis)
if axis is None:
a = np.ravel(a)
axis = 0
b2 = skew(a, axis)
n = float(a.shape[axis])
if n < 8:
raise ValueError(
"skewtest is not valid with less than 8 samples; %i samples"
" were given." % int(n))
y = b2 * math.sqrt(((n + 1) * (n + 3)) / (6.0 * (n - 2)))
beta2 = (3.0 * (n**2 + 27*n - 70) * (n+1) * (n+3) /
((n-2.0) * (n+5) * (n+7) * (n+9)))
W2 = -1 + math.sqrt(2 * (beta2 - 1))
delta = 1 / math.sqrt(0.5 * math.log(W2))
alpha = math.sqrt(2.0 / (W2 - 1))
y = np.where(y == 0, 1, y)
Z = delta * np.log(y / alpha + np.sqrt((y / alpha)**2 + 1))
return SkewtestResult(Z, 2 * distributions.norm.sf(np.abs(Z)))
KurtosistestResult = namedtuple('KurtosistestResult', ('statistic', 'pvalue'))
def kurtosistest(a, axis=0, nan_policy='propagate'):
"""
Test whether a dataset has normal kurtosis.
This function tests the null hypothesis that the kurtosis
of the population from which the sample was drawn is that
of the normal distribution: ``kurtosis = 3(n-1)/(n+1)``.
Parameters
----------
a : array
array of the sample data
axis : int or None, optional
Axis along which to compute test. Default is 0. If None,
compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float
The computed z-score for this test.
pvalue : float
The 2-sided p-value for the hypothesis test
Notes
-----
Valid only for n>20. The Z-score is set to 0 for bad entries.
This function uses the method described in [1]_.
References
----------
.. [1] see e.g. F. J. Anscombe, W. J. Glynn, "Distribution of the kurtosis
statistic b2 for normal samples", Biometrika, vol. 70, pp. 227-234, 1983.
Examples
--------
>>> from scipy.stats import kurtosistest
>>> kurtosistest(list(range(20)))
KurtosistestResult(statistic=-1.7058104152122062, pvalue=0.08804338332528348)
>>> np.random.seed(28041990)
>>> s = np.random.normal(0, 1, 1000)
>>> kurtosistest(s)
KurtosistestResult(statistic=1.2317590987707365, pvalue=0.21803908613450895)
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.kurtosistest(a, axis)
n = float(a.shape[axis])
if n < 5:
raise ValueError(
"kurtosistest requires at least 5 observations; %i observations"
" were given." % int(n))
if n < 20:
warnings.warn("kurtosistest only valid for n>=20 ... continuing "
"anyway, n=%i" % int(n))
b2 = kurtosis(a, axis, fisher=False)
E = 3.0*(n-1) / (n+1)
varb2 = 24.0*n*(n-2)*(n-3) / ((n+1)*(n+1.)*(n+3)*(n+5)) # [1]_ Eq. 1
x = (b2-E) / np.sqrt(varb2) # [1]_ Eq. 4
# [1]_ Eq. 2:
sqrtbeta1 = 6.0*(n*n-5*n+2)/((n+7)*(n+9)) * np.sqrt((6.0*(n+3)*(n+5)) /
(n*(n-2)*(n-3)))
# [1]_ Eq. 3:
A = 6.0 + 8.0/sqrtbeta1 * (2.0/sqrtbeta1 + np.sqrt(1+4.0/(sqrtbeta1**2)))
term1 = 1 - 2/(9.0*A)
denom = 1 + x*np.sqrt(2/(A-4.0))
denom = np.where(denom < 0, 99, denom)
term2 = np.where(denom < 0, term1, np.power((1-2.0/A)/denom, 1/3.0))
Z = (term1 - term2) / np.sqrt(2/(9.0*A)) # [1]_ Eq. 5
Z = np.where(denom == 99, 0, Z)
if Z.ndim == 0:
Z = Z[()]
# zprob uses upper tail, so Z needs to be positive
return KurtosistestResult(Z, 2 * distributions.norm.sf(np.abs(Z)))
NormaltestResult = namedtuple('NormaltestResult', ('statistic', 'pvalue'))
def normaltest(a, axis=0, nan_policy='propagate'):
"""
Test whether a sample differs from a normal distribution.
This function tests the null hypothesis that a sample comes
from a normal distribution. It is based on D'Agostino and
Pearson's [1]_, [2]_ test that combines skew and kurtosis to
produce an omnibus test of normality.
Parameters
----------
a : array_like
The array containing the sample to be tested.
axis : int or None, optional
Axis along which to compute test. Default is 0. If None,
compute over the whole array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float or array
``s^2 + k^2``, where ``s`` is the z-score returned by `skewtest` and
``k`` is the z-score returned by `kurtosistest`.
pvalue : float or array
A 2-sided chi squared probability for the hypothesis test.
References
----------
.. [1] D'Agostino, R. B. (1971), "An omnibus test of normality for
moderate and large sample size", Biometrika, 58, 341-348
.. [2] D'Agostino, R. and Pearson, E. S. (1973), "Tests for departure from
normality", Biometrika, 60, 613-622
Examples
--------
>>> from scipy import stats
>>> pts = 1000
>>> np.random.seed(28041990)
>>> a = np.random.normal(0, 1, size=pts)
>>> b = np.random.normal(2, 1, size=pts)
>>> x = np.concatenate((a, b))
>>> k2, p = stats.normaltest(x)
>>> alpha = 1e-3
>>> print("p = {:g}".format(p))
p = 3.27207e-11
>>> if p < alpha: # null hypothesis: x comes from a normal distribution
... print("The null hypothesis can be rejected")
... else:
... print("The null hypothesis cannot be rejected")
The null hypothesis can be rejected
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.normaltest(a, axis)
s, _ = skewtest(a, axis)
k, _ = kurtosistest(a, axis)
k2 = s*s + k*k
return NormaltestResult(k2, distributions.chi2.sf(k2, 2))
def jarque_bera(x):
"""
Perform the Jarque-Bera goodness of fit test on sample data.
The Jarque-Bera test tests whether the sample data has the skewness and
kurtosis matching a normal distribution.
Note that this test only works for a large enough number of data samples
(>2000) as the test statistic asymptotically has a Chi-squared distribution
with 2 degrees of freedom.
Parameters
----------
x : array_like
Observations of a random variable.
Returns
-------
jb_value : float
The test statistic.
p : float
The p-value for the hypothesis test.
References
----------
.. [1] Jarque, C. and Bera, A. (1980) "Efficient tests for normality,
homoscedasticity and serial independence of regression residuals",
6 Econometric Letters 255-259.
Examples
--------
>>> from scipy import stats
>>> np.random.seed(987654321)
>>> x = np.random.normal(0, 1, 100000)
>>> y = np.random.rayleigh(1, 100000)
>>> stats.jarque_bera(x)
(4.7165707989581342, 0.09458225503041906)
>>> stats.jarque_bera(y)
(6713.7098548143422, 0.0)
"""
x = np.asarray(x)
n = float(x.size)
if n == 0:
raise ValueError('At least one observation is required.')
mu = x.mean()
diffx = x - mu
skewness = (1 / n * np.sum(diffx**3)) / (1 / n * np.sum(diffx**2))**(3 / 2.)
kurtosis = (1 / n * np.sum(diffx**4)) / (1 / n * np.sum(diffx**2))**2
jb_value = n / 6 * (skewness**2 + (kurtosis - 3)**2 / 4)
p = 1 - distributions.chi2.cdf(jb_value, 2)
return jb_value, p
#####################################
# FREQUENCY FUNCTIONS #
#####################################
@np.deprecate(message="`itemfreq` is deprecated and will be removed in a "
"future version. Use instead `np.unique(..., return_counts=True)`")
def itemfreq(a):
"""
Return a 2-D array of item frequencies.
Parameters
----------
a : (N,) array_like
Input array.
Returns
-------
itemfreq : (K, 2) ndarray
A 2-D frequency table. Column 1 contains sorted, unique values from
`a`, column 2 contains their respective counts.
Examples
--------
>>> from scipy import stats
>>> a = np.array([1, 1, 5, 0, 1, 2, 2, 0, 1, 4])
>>> stats.itemfreq(a)
array([[ 0., 2.],
[ 1., 4.],
[ 2., 2.],
[ 4., 1.],
[ 5., 1.]])
>>> np.bincount(a)
array([2, 4, 2, 0, 1, 1])
>>> stats.itemfreq(a/10.)
array([[ 0. , 2. ],
[ 0.1, 4. ],
[ 0.2, 2. ],
[ 0.4, 1. ],
[ 0.5, 1. ]])
"""
items, inv = np.unique(a, return_inverse=True)
freq = np.bincount(inv)
return np.array([items, freq]).T
def scoreatpercentile(a, per, limit=(), interpolation_method='fraction',
axis=None):
"""
Calculate the score at a given percentile of the input sequence.
For example, the score at `per=50` is the median. If the desired quantile
lies between two data points, we interpolate between them, according to
the value of `interpolation`. If the parameter `limit` is provided, it
should be a tuple (lower, upper) of two values.
Parameters
----------
a : array_like
A 1-D array of values from which to extract score.
per : array_like
Percentile(s) at which to extract score. Values should be in range
[0,100].
limit : tuple, optional
Tuple of two scalars, the lower and upper limits within which to
compute the percentile. Values of `a` outside
this (closed) interval will be ignored.
interpolation_method : {'fraction', 'lower', 'higher'}, optional
This optional parameter specifies the interpolation method to use,
when the desired quantile lies between two data points `i` and `j`
- fraction: ``i + (j - i) * fraction`` where ``fraction`` is the
fractional part of the index surrounded by ``i`` and ``j``.
- lower: ``i``.
- higher: ``j``.
axis : int, optional
Axis along which the percentiles are computed. Default is None. If
None, compute over the whole array `a`.
Returns
-------
score : float or ndarray
Score at percentile(s).
See Also
--------
percentileofscore, numpy.percentile
Notes
-----
This function will become obsolete in the future.
For Numpy 1.9 and higher, `numpy.percentile` provides all the functionality
that `scoreatpercentile` provides. And it's significantly faster.
Therefore it's recommended to use `numpy.percentile` for users that have
numpy >= 1.9.
Examples
--------
>>> from scipy import stats
>>> a = np.arange(100)
>>> stats.scoreatpercentile(a, 50)
49.5
"""
# adapted from NumPy's percentile function. When we require numpy >= 1.8,
# the implementation of this function can be replaced by np.percentile.
a = np.asarray(a)
if a.size == 0:
# empty array, return nan(s) with shape matching `per`
if np.isscalar(per):
return np.nan
else:
return np.ones(np.asarray(per).shape, dtype=np.float64) * np.nan
if limit:
a = a[(limit[0] <= a) & (a <= limit[1])]
sorted = np.sort(a, axis=axis)
if axis is None:
axis = 0
return _compute_qth_percentile(sorted, per, interpolation_method, axis)
# handle sequence of per's without calling sort multiple times
def _compute_qth_percentile(sorted, per, interpolation_method, axis):
if not np.isscalar(per):
score = [_compute_qth_percentile(sorted, i, interpolation_method, axis)
for i in per]
return np.array(score)
if (per < 0) or (per > 100):
raise ValueError("percentile must be in the range [0, 100]")
indexer = [slice(None)] * sorted.ndim
idx = per / 100. * (sorted.shape[axis] - 1)
if int(idx) != idx:
# round fractional indices according to interpolation method
if interpolation_method == 'lower':
idx = int(np.floor(idx))
elif interpolation_method == 'higher':
idx = int(np.ceil(idx))
elif interpolation_method == 'fraction':
pass # keep idx as fraction and interpolate
else:
raise ValueError("interpolation_method can only be 'fraction', "
"'lower' or 'higher'")
i = int(idx)
if i == idx:
indexer[axis] = slice(i, i + 1)
weights = array(1)
sumval = 1.0
else:
indexer[axis] = slice(i, i + 2)
j = i + 1
weights = array([(j - idx), (idx - i)], float)
wshape = [1] * sorted.ndim
wshape[axis] = 2
weights.shape = wshape
sumval = weights.sum()
# Use np.add.reduce (== np.sum but a little faster) to coerce data type
return np.add.reduce(sorted[indexer] * weights, axis=axis) / sumval
def percentileofscore(a, score, kind='rank'):
"""
The percentile rank of a score relative to a list of scores.
A `percentileofscore` of, for example, 80% means that 80% of the
scores in `a` are below the given score. In the case of gaps or
ties, the exact definition depends on the optional keyword, `kind`.
Parameters
----------
a : array_like
Array of scores to which `score` is compared.
score : int or float
Score that is compared to the elements in `a`.
kind : {'rank', 'weak', 'strict', 'mean'}, optional
This optional parameter specifies the interpretation of the
resulting score:
- "rank": Average percentage ranking of score. In case of
multiple matches, average the percentage rankings of
all matching scores.
- "weak": This kind corresponds to the definition of a cumulative
distribution function. A percentileofscore of 80%
means that 80% of values are less than or equal
to the provided score.
- "strict": Similar to "weak", except that only values that are
strictly less than the given score are counted.
- "mean": The average of the "weak" and "strict" scores, often used in
testing. See
http://en.wikipedia.org/wiki/Percentile_rank
Returns
-------
pcos : float
Percentile-position of score (0-100) relative to `a`.
See Also
--------
numpy.percentile
Examples
--------
Three-quarters of the given values lie below a given score:
>>> from scipy import stats
>>> stats.percentileofscore([1, 2, 3, 4], 3)
75.0
With multiple matches, note how the scores of the two matches, 0.6
and 0.8 respectively, are averaged:
>>> stats.percentileofscore([1, 2, 3, 3, 4], 3)
70.0
Only 2/5 values are strictly less than 3:
>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='strict')
40.0
But 4/5 values are less than or equal to 3:
>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='weak')
80.0
The average between the weak and the strict scores is
>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='mean')
60.0
"""
if np.isnan(score):
return np.nan
a = np.asarray(a)
n = len(a)
if n == 0:
return 100.0
if kind == 'rank':
left = np.count_nonzero(a < score)
right = np.count_nonzero(a <= score)
pct = (right + left + (1 if right > left else 0)) * 50.0/n
return pct
elif kind == 'strict':
return np.count_nonzero(a < score) / float(n) * 100
elif kind == 'weak':
return np.count_nonzero(a <= score) / float(n) * 100
elif kind == 'mean':
pct = (np.count_nonzero(a < score) + np.count_nonzero(a <= score)) / float(n) * 50
return pct
else:
raise ValueError("kind can only be 'rank', 'strict', 'weak' or 'mean'")
HistogramResult = namedtuple('HistogramResult',
('count', 'lowerlimit', 'binsize', 'extrapoints'))
def _histogram(a, numbins=10, defaultlimits=None, weights=None, printextras=False):
"""
Separate the range into several bins and return the number of instances
in each bin.
Parameters
----------
a : array_like
Array of scores which will be put into bins.
numbins : int, optional
The number of bins to use for the histogram. Default is 10.
defaultlimits : tuple (lower, upper), optional
The lower and upper values for the range of the histogram.
If no value is given, a range slightly larger than the range of the
values in a is used. Specifically ``(a.min() - s, a.max() + s)``,
where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``.
weights : array_like, optional
The weights for each value in `a`. Default is None, which gives each
value a weight of 1.0
printextras : bool, optional
If True, if there are extra points (i.e. the points that fall outside
the bin limits) a warning is raised saying how many of those points
there are. Default is False.
Returns
-------
count : ndarray
Number of points (or sum of weights) in each bin.
lowerlimit : float
Lowest value of histogram, the lower limit of the first bin.
binsize : float
The size of the bins (all bins have the same size).
extrapoints : int
The number of points outside the range of the histogram.
See Also
--------
numpy.histogram
Notes
-----
This histogram is based on numpy's histogram but has a larger range by
default if default limits is not set.
"""
a = np.ravel(a)
if defaultlimits is None:
if a.size == 0:
# handle empty arrays. Undetermined range, so use 0-1.
defaultlimits = (0, 1)
else:
# no range given, so use values in `a`
data_min = a.min()
data_max = a.max()
# Have bins extend past min and max values slightly
s = (data_max - data_min) / (2. * (numbins - 1.))
defaultlimits = (data_min - s, data_max + s)
# use numpy's histogram method to compute bins
hist, bin_edges = np.histogram(a, bins=numbins, range=defaultlimits,
weights=weights)
# hist are not always floats, convert to keep with old output
hist = np.array(hist, dtype=float)
# fixed width for bins is assumed, as numpy's histogram gives
# fixed width bins for int values for 'bins'
binsize = bin_edges[1] - bin_edges[0]
# calculate number of extra points
extrapoints = len([v for v in a
if defaultlimits[0] > v or v > defaultlimits[1]])
if extrapoints > 0 and printextras:
warnings.warn("Points outside given histogram range = %s"
% extrapoints)
return HistogramResult(hist, defaultlimits[0], binsize, extrapoints)
CumfreqResult = namedtuple('CumfreqResult',
('cumcount', 'lowerlimit', 'binsize',
'extrapoints'))
def cumfreq(a, numbins=10, defaultreallimits=None, weights=None):
"""
Return a cumulative frequency histogram, using the histogram function.
A cumulative histogram is a mapping that counts the cumulative number of
observations in all of the bins up to the specified bin.
Parameters
----------
a : array_like
Input array.
numbins : int, optional
The number of bins to use for the histogram. Default is 10.
defaultreallimits : tuple (lower, upper), optional
The lower and upper values for the range of the histogram.
If no value is given, a range slightly larger than the range of the
values in `a` is used. Specifically ``(a.min() - s, a.max() + s)``,
where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``.
weights : array_like, optional
The weights for each value in `a`. Default is None, which gives each
value a weight of 1.0
Returns
-------
cumcount : ndarray
Binned values of cumulative frequency.
lowerlimit : float
Lower real limit
binsize : float
Width of each bin.
extrapoints : int
Extra points.
Examples
--------
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> x = [1, 4, 2, 1, 3, 1]
>>> res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5))
>>> res.cumcount
array([ 1., 2., 3., 3.])
>>> res.extrapoints
3
Create a normal distribution with 1000 random values
>>> rng = np.random.RandomState(seed=12345)
>>> samples = stats.norm.rvs(size=1000, random_state=rng)
Calculate cumulative frequencies
>>> res = stats.cumfreq(samples, numbins=25)
Calculate space of values for x
>>> x = res.lowerlimit + np.linspace(0, res.binsize*res.cumcount.size,
... res.cumcount.size)
Plot histogram and cumulative histogram
>>> fig = plt.figure(figsize=(10, 4))
>>> ax1 = fig.add_subplot(1, 2, 1)
>>> ax2 = fig.add_subplot(1, 2, 2)
>>> ax1.hist(samples, bins=25)
>>> ax1.set_title('Histogram')
>>> ax2.bar(x, res.cumcount, width=res.binsize)
>>> ax2.set_title('Cumulative histogram')
>>> ax2.set_xlim([x.min(), x.max()])
>>> plt.show()
"""
h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights)
cumhist = np.cumsum(h * 1, axis=0)
return CumfreqResult(cumhist, l, b, e)
RelfreqResult = namedtuple('RelfreqResult',
('frequency', 'lowerlimit', 'binsize',
'extrapoints'))
def relfreq(a, numbins=10, defaultreallimits=None, weights=None):
"""
Return a relative frequency histogram, using the histogram function.
A relative frequency histogram is a mapping of the number of
observations in each of the bins relative to the total of observations.
Parameters
----------
a : array_like
Input array.
numbins : int, optional
The number of bins to use for the histogram. Default is 10.
defaultreallimits : tuple (lower, upper), optional
The lower and upper values for the range of the histogram.
If no value is given, a range slightly larger than the range of the
values in a is used. Specifically ``(a.min() - s, a.max() + s)``,
where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``.
weights : array_like, optional
The weights for each value in `a`. Default is None, which gives each
value a weight of 1.0
Returns
-------
frequency : ndarray
Binned values of relative frequency.
lowerlimit : float
Lower real limit
binsize : float
Width of each bin.
extrapoints : int
Extra points.
Examples
--------
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> a = np.array([2, 4, 1, 2, 3, 2])
>>> res = stats.relfreq(a, numbins=4)
>>> res.frequency
array([ 0.16666667, 0.5 , 0.16666667, 0.16666667])
>>> np.sum(res.frequency) # relative frequencies should add up to 1
1.0
Create a normal distribution with 1000 random values
>>> rng = np.random.RandomState(seed=12345)
>>> samples = stats.norm.rvs(size=1000, random_state=rng)
Calculate relative frequencies
>>> res = stats.relfreq(samples, numbins=25)
Calculate space of values for x
>>> x = res.lowerlimit + np.linspace(0, res.binsize*res.frequency.size,
... res.frequency.size)
Plot relative frequency histogram
>>> fig = plt.figure(figsize=(5, 4))
>>> ax = fig.add_subplot(1, 1, 1)
>>> ax.bar(x, res.frequency, width=res.binsize)
>>> ax.set_title('Relative frequency histogram')
>>> ax.set_xlim([x.min(), x.max()])
>>> plt.show()
"""
a = np.asanyarray(a)
h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights)
h = h / float(a.shape[0])
return RelfreqResult(h, l, b, e)
#####################################
# VARIABILITY FUNCTIONS #
#####################################
def obrientransform(*args):
"""
Compute the O'Brien transform on input data (any number of arrays).
Used to test for homogeneity of variance prior to running one-way stats.
Each array in ``*args`` is one level of a factor.
If `f_oneway` is run on the transformed data and found significant,
the variances are unequal. From Maxwell and Delaney [1]_, p.112.
Parameters
----------
args : tuple of array_like
Any number of arrays.
Returns
-------
obrientransform : ndarray
Transformed data for use in an ANOVA. The first dimension
of the result corresponds to the sequence of transformed
arrays. If the arrays given are all 1-D of the same length,
the return value is a 2-D array; otherwise it is a 1-D array
of type object, with each element being an ndarray.
References
----------
.. [1] S. E. Maxwell and H. D. Delaney, "Designing Experiments and
Analyzing Data: A Model Comparison Perspective", Wadsworth, 1990.
Examples
--------
We'll test the following data sets for differences in their variance.
>>> x = [10, 11, 13, 9, 7, 12, 12, 9, 10]
>>> y = [13, 21, 5, 10, 8, 14, 10, 12, 7, 15]
Apply the O'Brien transform to the data.
>>> from scipy.stats import obrientransform
>>> tx, ty = obrientransform(x, y)
Use `scipy.stats.f_oneway` to apply a one-way ANOVA test to the
transformed data.
>>> from scipy.stats import f_oneway
>>> F, p = f_oneway(tx, ty)
>>> p
0.1314139477040335
If we require that ``p < 0.05`` for significance, we cannot conclude
that the variances are different.
"""
TINY = np.sqrt(np.finfo(float).eps)
# `arrays` will hold the transformed arguments.
arrays = []
for arg in args:
a = np.asarray(arg)
n = len(a)
mu = np.mean(a)
sq = (a - mu)**2
sumsq = sq.sum()
# The O'Brien transform.
t = ((n - 1.5) * n * sq - 0.5 * sumsq) / ((n - 1) * (n - 2))
# Check that the mean of the transformed data is equal to the
# original variance.
var = sumsq / (n - 1)
if abs(var - np.mean(t)) > TINY:
raise ValueError('Lack of convergence in obrientransform.')
arrays.append(t)
return np.array(arrays)
def sem(a, axis=0, ddof=1, nan_policy='propagate'):
"""
Calculate the standard error of the mean (or standard error of
measurement) of the values in the input array.
Parameters
----------
a : array_like
An array containing the values for which the standard error is
returned.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over
the whole array `a`.
ddof : int, optional
Delta degrees-of-freedom. How many degrees of freedom to adjust
for bias in limited samples relative to the population estimate
of variance. Defaults to 1.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
s : ndarray or float
The standard error of the mean in the sample(s), along the input axis.
Notes
-----
The default value for `ddof` is different to the default (0) used by other
ddof containing routines, such as np.std and np.nanstd.
Examples
--------
Find standard error along the first axis:
>>> from scipy import stats
>>> a = np.arange(20).reshape(5,4)
>>> stats.sem(a)
array([ 2.8284, 2.8284, 2.8284, 2.8284])
Find standard error across the whole array, using n degrees of freedom:
>>> stats.sem(a, axis=None, ddof=0)
1.2893796958227628
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.sem(a, axis, ddof)
n = a.shape[axis]
s = np.std(a, axis=axis, ddof=ddof) / np.sqrt(n)
return s
def zscore(a, axis=0, ddof=0):
"""
Calculate the z score of each value in the sample, relative to the
sample mean and standard deviation.
Parameters
----------
a : array_like
An array like object containing the sample data.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over
the whole array `a`.
ddof : int, optional
Degrees of freedom correction in the calculation of the
standard deviation. Default is 0.
Returns
-------
zscore : array_like
The z-scores, standardized by mean and standard deviation of
input array `a`.
Notes
-----
This function preserves ndarray subclasses, and works also with
matrices and masked arrays (it uses `asanyarray` instead of
`asarray` for parameters).
Examples
--------
>>> a = np.array([ 0.7972, 0.0767, 0.4383, 0.7866, 0.8091,
... 0.1954, 0.6307, 0.6599, 0.1065, 0.0508])
>>> from scipy import stats
>>> stats.zscore(a)
array([ 1.1273, -1.247 , -0.0552, 1.0923, 1.1664, -0.8559, 0.5786,
0.6748, -1.1488, -1.3324])
Computing along a specified axis, using n-1 degrees of freedom
(``ddof=1``) to calculate the standard deviation:
>>> b = np.array([[ 0.3148, 0.0478, 0.6243, 0.4608],
... [ 0.7149, 0.0775, 0.6072, 0.9656],
... [ 0.6341, 0.1403, 0.9759, 0.4064],
... [ 0.5918, 0.6948, 0.904 , 0.3721],
... [ 0.0921, 0.2481, 0.1188, 0.1366]])
>>> stats.zscore(b, axis=1, ddof=1)
array([[-0.19264823, -1.28415119, 1.07259584, 0.40420358],
[ 0.33048416, -1.37380874, 0.04251374, 1.00081084],
[ 0.26796377, -1.12598418, 1.23283094, -0.37481053],
[-0.22095197, 0.24468594, 1.19042819, -1.21416216],
[-0.82780366, 1.4457416 , -0.43867764, -0.1792603 ]])
"""
a = np.asanyarray(a)
mns = a.mean(axis=axis)
sstd = a.std(axis=axis, ddof=ddof)
if axis and mns.ndim < a.ndim:
return ((a - np.expand_dims(mns, axis=axis)) /
np.expand_dims(sstd, axis=axis))
else:
return (a - mns) / sstd
def zmap(scores, compare, axis=0, ddof=0):
"""
Calculate the relative z-scores.
Return an array of z-scores, i.e., scores that are standardized to
zero mean and unit variance, where mean and variance are calculated
from the comparison array.
Parameters
----------
scores : array_like
The input for which z-scores are calculated.
compare : array_like
The input from which the mean and standard deviation of the
normalization are taken; assumed to have the same dimension as
`scores`.
axis : int or None, optional
Axis over which mean and variance of `compare` are calculated.
Default is 0. If None, compute over the whole array `scores`.
ddof : int, optional
Degrees of freedom correction in the calculation of the
standard deviation. Default is 0.
Returns
-------
zscore : array_like
Z-scores, in the same shape as `scores`.
Notes
-----
This function preserves ndarray subclasses, and works also with
matrices and masked arrays (it uses `asanyarray` instead of
`asarray` for parameters).
Examples
--------
>>> from scipy.stats import zmap
>>> a = [0.5, 2.0, 2.5, 3]
>>> b = [0, 1, 2, 3, 4]
>>> zmap(a, b)
array([-1.06066017, 0. , 0.35355339, 0.70710678])
"""
scores, compare = map(np.asanyarray, [scores, compare])
mns = compare.mean(axis=axis)
sstd = compare.std(axis=axis, ddof=ddof)
if axis and mns.ndim < compare.ndim:
return ((scores - np.expand_dims(mns, axis=axis)) /
np.expand_dims(sstd, axis=axis))
else:
return (scores - mns) / sstd
# Private dictionary initialized only once at module level
# See https://en.wikipedia.org/wiki/Robust_measures_of_scale
_scale_conversions = {'raw': 1.0,
'normal': special.erfinv(0.5) * 2.0 * math.sqrt(2.0)}
def iqr(x, axis=None, rng=(25, 75), scale='raw', nan_policy='propagate',
interpolation='linear', keepdims=False):
"""
Compute the interquartile range of the data along the specified axis.
The interquartile range (IQR) is the difference between the 75th and
25th percentile of the data. It is a measure of the dispersion
similar to standard deviation or variance, but is much more robust
against outliers [2]_.
The ``rng`` parameter allows this function to compute other
percentile ranges than the actual IQR. For example, setting
``rng=(0, 100)`` is equivalent to `numpy.ptp`.
The IQR of an empty array is `np.nan`.
.. versionadded:: 0.18.0
Parameters
----------
x : array_like
Input array or object that can be converted to an array.
axis : int or sequence of int, optional
Axis along which the range is computed. The default is to
compute the IQR for the entire array.
rng : Two-element sequence containing floats in range of [0,100] optional
Percentiles over which to compute the range. Each must be
between 0 and 100, inclusive. The default is the true IQR:
`(25, 75)`. The order of the elements is not important.
scale : scalar or str, optional
The numerical value of scale will be divided out of the final
result. The following string values are recognized:
'raw' : No scaling, just return the raw IQR.
'normal' : Scale by :math:`2 \\sqrt{2} erf^{-1}(\\frac{1}{2}) \\approx 1.349`.
The default is 'raw'. Array-like scale is also allowed, as long
as it broadcasts correctly to the output such that
``out / scale`` is a valid operation. The output dimensions
depend on the input array, `x`, the `axis` argument, and the
`keepdims` flag.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate'
returns nan, 'raise' throws an error, 'omit' performs the
calculations ignoring nan values. Default is 'propagate'.
interpolation : {'linear', 'lower', 'higher', 'midpoint', 'nearest'}, optional
Specifies the interpolation method to use when the percentile
boundaries lie between two data points `i` and `j`:
* 'linear' : `i + (j - i) * fraction`, where `fraction` is the
fractional part of the index surrounded by `i` and `j`.
* 'lower' : `i`.
* 'higher' : `j`.
* 'nearest' : `i` or `j` whichever is nearest.
* 'midpoint' : `(i + j) / 2`.
Default is 'linear'.
keepdims : bool, optional
If this is set to `True`, the reduced axes are left in the
result as dimensions with size one. With this option, the result
will broadcast correctly against the original array `x`.
Returns
-------
iqr : scalar or ndarray
If ``axis=None``, a scalar is returned. If the input contains
integers or floats of smaller precision than ``np.float64``, then the
output data-type is ``np.float64``. Otherwise, the output data-type is
the same as that of the input.
See Also
--------
numpy.std, numpy.var
Examples
--------
>>> from scipy.stats import iqr
>>> x = np.array([[10, 7, 4], [3, 2, 1]])
>>> x
array([[10, 7, 4],
[ 3, 2, 1]])
>>> iqr(x)
4.0
>>> iqr(x, axis=0)
array([ 3.5, 2.5, 1.5])
>>> iqr(x, axis=1)
array([ 3., 1.])
>>> iqr(x, axis=1, keepdims=True)
array([[ 3.],
[ 1.]])
Notes
-----
This function is heavily dependent on the version of `numpy` that is
installed. Versions greater than 1.11.0b3 are highly recommended, as they
include a number of enhancements and fixes to `numpy.percentile` and
`numpy.nanpercentile` that affect the operation of this function. The
following modifications apply:
Below 1.10.0 : `nan_policy` is poorly defined.
The default behavior of `numpy.percentile` is used for 'propagate'. This
is a hybrid of 'omit' and 'propagate' that mostly yields a skewed
version of 'omit' since NaNs are sorted to the end of the data. A
warning is raised if there are NaNs in the data.
Below 1.9.0: `numpy.nanpercentile` does not exist.
This means that `numpy.percentile` is used regardless of `nan_policy`
and a warning is issued. See previous item for a description of the
behavior.
Below 1.9.0: `keepdims` and `interpolation` are not supported.
The keywords get ignored with a warning if supplied with non-default
values. However, multiple axes are still supported.
References
----------
.. [1] "Interquartile range" https://en.wikipedia.org/wiki/Interquartile_range
.. [2] "Robust measures of scale" https://en.wikipedia.org/wiki/Robust_measures_of_scale
.. [3] "Quantile" https://en.wikipedia.org/wiki/Quantile
"""
x = asarray(x)
# This check prevents percentile from raising an error later. Also, it is
# consistent with `np.var` and `np.std`.
if not x.size:
return np.nan
# An error may be raised here, so fail-fast, before doing lengthy
# computations, even though `scale` is not used until later
if isinstance(scale, string_types):
scale_key = scale.lower()
if scale_key not in _scale_conversions:
raise ValueError("{0} not a valid scale for `iqr`".format(scale))
scale = _scale_conversions[scale_key]
# Select the percentile function to use based on nans and policy
contains_nan, nan_policy = _contains_nan(x, nan_policy)
if contains_nan and nan_policy == 'omit':
percentile_func = _iqr_nanpercentile
else:
percentile_func = _iqr_percentile
if len(rng) != 2:
raise TypeError("quantile range must be two element sequence")
rng = sorted(rng)
pct = percentile_func(x, rng, axis=axis, interpolation=interpolation,
keepdims=keepdims, contains_nan=contains_nan)
out = np.subtract(pct[1], pct[0])
if scale != 1.0:
out /= scale
return out
def _iqr_percentile(x, q, axis=None, interpolation='linear', keepdims=False, contains_nan=False):
"""
Private wrapper that works around older versions of `numpy`.
While this function is pretty much necessary for the moment, it
should be removed as soon as the minimum supported numpy version
allows.
"""
if contains_nan and NumpyVersion(np.__version__) < '1.10.0a':
# I see no way to avoid the version check to ensure that the corrected
# NaN behavior has been implemented except to call `percentile` on a
# small array.
msg = "Keyword nan_policy='propagate' not correctly supported for " \
"numpy versions < 1.10.x. The default behavior of " \
"`numpy.percentile` will be used."
warnings.warn(msg, RuntimeWarning)
try:
# For older versions of numpy, there are two things that can cause a
# problem here: missing keywords and non-scalar axis. The former can be
# partially handled with a warning, the latter can be handled fully by
# hacking in an implementation similar to numpy's function for
# providing multi-axis functionality
# (`numpy.lib.function_base._ureduce` for the curious).
result = np.percentile(x, q, axis=axis, keepdims=keepdims,
interpolation=interpolation)
except TypeError:
if interpolation != 'linear' or keepdims:
# At time or writing, this means np.__version__ < 1.9.0
warnings.warn("Keywords interpolation and keepdims not supported "
"for your version of numpy", RuntimeWarning)
try:
# Special processing if axis is an iterable
original_size = len(axis)
except TypeError:
# Axis is a scalar at this point
pass
else:
axis = np.unique(np.asarray(axis) % x.ndim)
if original_size > axis.size:
# mimic numpy if axes are duplicated
raise ValueError("duplicate value in axis")
if axis.size == x.ndim:
# axis includes all axes: revert to None
axis = None
elif axis.size == 1:
# no rolling necessary
axis = axis[0]
else:
# roll multiple axes to the end and flatten that part out
for ax in axis[::-1]:
x = np.rollaxis(x, ax, x.ndim)
x = x.reshape(x.shape[:-axis.size] +
(np.prod(x.shape[-axis.size:]),))
axis = -1
result = np.percentile(x, q, axis=axis)
return result
def _iqr_nanpercentile(x, q, axis=None, interpolation='linear', keepdims=False,
contains_nan=False):
"""
Private wrapper that works around the following:
1. A bug in `np.nanpercentile` that was around until numpy version
1.11.0.
2. A bug in `np.percentile` NaN handling that was fixed in numpy
version 1.10.0.
3. The non-existence of `np.nanpercentile` before numpy version
1.9.0.
While this function is pretty much necessary for the moment, it
should be removed as soon as the minimum supported numpy version
allows.
"""
if hasattr(np, 'nanpercentile'):
# At time or writing, this means np.__version__ < 1.9.0
result = np.nanpercentile(x, q, axis=axis,
interpolation=interpolation,
keepdims=keepdims)
# If non-scalar result and nanpercentile does not do proper axis roll.
# I see no way of avoiding the version test since dimensions may just
# happen to match in the data.
if result.ndim > 1 and NumpyVersion(np.__version__) < '1.11.0a':
axis = np.asarray(axis)
if axis.size == 1:
# If only one axis specified, reduction happens along that dimension
if axis.ndim == 0:
axis = axis[None]
result = np.rollaxis(result, axis[0])
else:
# If multiple axes, reduced dimeision is last
result = np.rollaxis(result, -1)
else:
msg = "Keyword nan_policy='omit' not correctly supported for numpy " \
"versions < 1.9.x. The default behavior of numpy.percentile " \
"will be used."
warnings.warn(msg, RuntimeWarning)
result = _iqr_percentile(x, q, axis=axis)
return result
#####################################
# TRIMMING FUNCTIONS #
#####################################
SigmaclipResult = namedtuple('SigmaclipResult', ('clipped', 'lower', 'upper'))
def sigmaclip(a, low=4., high=4.):
"""
Iterative sigma-clipping of array elements.
Starting from the full sample, all elements outside the critical range are
removed, i.e. all elements of the input array `c` that satisfy either of
the following conditions ::
c < mean(c) - std(c)*low
c > mean(c) + std(c)*high
The iteration continues with the updated sample until no
elements are outside the (updated) range.
Parameters
----------
a : array_like
Data array, will be raveled if not 1-D.
low : float, optional
Lower bound factor of sigma clipping. Default is 4.
high : float, optional
Upper bound factor of sigma clipping. Default is 4.
Returns
-------
clipped : ndarray
Input array with clipped elements removed.
lower : float
Lower threshold value use for clipping.
upper : float
Upper threshold value use for clipping.
Examples
--------
>>> from scipy.stats import sigmaclip
>>> a = np.concatenate((np.linspace(9.5, 10.5, 31),
... np.linspace(0, 20, 5)))
>>> fact = 1.5
>>> c, low, upp = sigmaclip(a, fact, fact)
>>> c
array([ 9.96666667, 10. , 10.03333333, 10. ])
>>> c.var(), c.std()
(0.00055555555555555165, 0.023570226039551501)
>>> low, c.mean() - fact*c.std(), c.min()
(9.9646446609406727, 9.9646446609406727, 9.9666666666666668)
>>> upp, c.mean() + fact*c.std(), c.max()
(10.035355339059327, 10.035355339059327, 10.033333333333333)
>>> a = np.concatenate((np.linspace(9.5, 10.5, 11),
... np.linspace(-100, -50, 3)))
>>> c, low, upp = sigmaclip(a, 1.8, 1.8)
>>> (c == np.linspace(9.5, 10.5, 11)).all()
True
"""
c = np.asarray(a).ravel()
delta = 1
while delta:
c_std = c.std()
c_mean = c.mean()
size = c.size
critlower = c_mean - c_std * low
critupper = c_mean + c_std * high
c = c[(c >= critlower) & (c <= critupper)]
delta = size - c.size
return SigmaclipResult(c, critlower, critupper)
def trimboth(a, proportiontocut, axis=0):
"""
Slices off a proportion of items from both ends of an array.
Slices off the passed proportion of items from both ends of the passed
array (i.e., with `proportiontocut` = 0.1, slices leftmost 10% **and**
rightmost 10% of scores). The trimmed values are the lowest and
highest ones.
Slices off less if proportion results in a non-integer slice index (i.e.,
conservatively slices off`proportiontocut`).
Parameters
----------
a : array_like
Data to trim.
proportiontocut : float
Proportion (in range 0-1) of total data set to trim of each end.
axis : int or None, optional
Axis along which to trim data. Default is 0. If None, compute over
the whole array `a`.
Returns
-------
out : ndarray
Trimmed version of array `a`. The order of the trimmed content
is undefined.
See Also
--------
trim_mean
Examples
--------
>>> from scipy import stats
>>> a = np.arange(20)
>>> b = stats.trimboth(a, 0.1)
>>> b.shape
(16,)
"""
a = np.asarray(a)
if a.size == 0:
return a
if axis is None:
a = a.ravel()
axis = 0
nobs = a.shape[axis]
lowercut = int(proportiontocut * nobs)
uppercut = nobs - lowercut
if (lowercut >= uppercut):
raise ValueError("Proportion too big.")
atmp = np.partition(a, (lowercut, uppercut - 1), axis)
sl = [slice(None)] * atmp.ndim
sl[axis] = slice(lowercut, uppercut)
return atmp[sl]
def trim1(a, proportiontocut, tail='right', axis=0):
"""
Slices off a proportion from ONE end of the passed array distribution.
If `proportiontocut` = 0.1, slices off 'leftmost' or 'rightmost'
10% of scores. The lowest or highest values are trimmed (depending on
the tail).
Slices off less if proportion results in a non-integer slice index
(i.e., conservatively slices off `proportiontocut` ).
Parameters
----------
a : array_like
Input array
proportiontocut : float
Fraction to cut off of 'left' or 'right' of distribution
tail : {'left', 'right'}, optional
Defaults to 'right'.
axis : int or None, optional
Axis along which to trim data. Default is 0. If None, compute over
the whole array `a`.
Returns
-------
trim1 : ndarray
Trimmed version of array `a`. The order of the trimmed content is
undefined.
"""
a = np.asarray(a)
if axis is None:
a = a.ravel()
axis = 0
nobs = a.shape[axis]
# avoid possible corner case
if proportiontocut >= 1:
return []
if tail.lower() == 'right':
lowercut = 0
uppercut = nobs - int(proportiontocut * nobs)
elif tail.lower() == 'left':
lowercut = int(proportiontocut * nobs)
uppercut = nobs
atmp = np.partition(a, (lowercut, uppercut - 1), axis)
return atmp[lowercut:uppercut]
def trim_mean(a, proportiontocut, axis=0):
"""
Return mean of array after trimming distribution from both tails.
If `proportiontocut` = 0.1, slices off 'leftmost' and 'rightmost' 10% of
scores. The input is sorted before slicing. Slices off less if proportion
results in a non-integer slice index (i.e., conservatively slices off
`proportiontocut` ).
Parameters
----------
a : array_like
Input array
proportiontocut : float
Fraction to cut off of both tails of the distribution
axis : int or None, optional
Axis along which the trimmed means are computed. Default is 0.
If None, compute over the whole array `a`.
Returns
-------
trim_mean : ndarray
Mean of trimmed array.
See Also
--------
trimboth
tmean : compute the trimmed mean ignoring values outside given `limits`.
Examples
--------
>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.trim_mean(x, 0.1)
9.5
>>> x2 = x.reshape(5, 4)
>>> x2
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15],
[16, 17, 18, 19]])
>>> stats.trim_mean(x2, 0.25)
array([ 8., 9., 10., 11.])
>>> stats.trim_mean(x2, 0.25, axis=1)
array([ 1.5, 5.5, 9.5, 13.5, 17.5])
"""
a = np.asarray(a)
if a.size == 0:
return np.nan
if axis is None:
a = a.ravel()
axis = 0
nobs = a.shape[axis]
lowercut = int(proportiontocut * nobs)
uppercut = nobs - lowercut
if (lowercut > uppercut):
raise ValueError("Proportion too big.")
atmp = np.partition(a, (lowercut, uppercut - 1), axis)
sl = [slice(None)] * atmp.ndim
sl[axis] = slice(lowercut, uppercut)
return np.mean(atmp[sl], axis=axis)
F_onewayResult = namedtuple('F_onewayResult', ('statistic', 'pvalue'))
def f_oneway(*args):
"""
Performs a 1-way ANOVA.
The one-way ANOVA tests the null hypothesis that two or more groups have
the same population mean. The test is applied to samples from two or
more groups, possibly with differing sizes.
Parameters
----------
sample1, sample2, ... : array_like
The sample measurements for each group.
Returns
-------
statistic : float
The computed F-value of the test.
pvalue : float
The associated p-value from the F-distribution.
Notes
-----
The ANOVA test has important assumptions that must be satisfied in order
for the associated p-value to be valid.
1. The samples are independent.
2. Each sample is from a normally distributed population.
3. The population standard deviations of the groups are all equal. This
property is known as homoscedasticity.
If these assumptions are not true for a given set of data, it may still be
possible to use the Kruskal-Wallis H-test (`scipy.stats.kruskal`) although
with some loss of power.
The algorithm is from Heiman[2], pp.394-7.
References
----------
.. [1] Lowry, Richard. "Concepts and Applications of Inferential
Statistics". Chapter 14.
http://faculty.vassar.edu/lowry/ch14pt1.html
.. [2] Heiman, G.W. Research Methods in Statistics. 2002.
.. [3] McDonald, G. H. "Handbook of Biological Statistics", One-way ANOVA.
http://www.biostathandbook.com/onewayanova.html
Examples
--------
>>> import scipy.stats as stats
[3]_ Here are some data on a shell measurement (the length of the anterior
adductor muscle scar, standardized by dividing by length) in the mussel
Mytilus trossulus from five locations: Tillamook, Oregon; Newport, Oregon;
Petersburg, Alaska; Magadan, Russia; and Tvarminne, Finland, taken from a
much larger data set used in McDonald et al. (1991).
>>> tillamook = [0.0571, 0.0813, 0.0831, 0.0976, 0.0817, 0.0859, 0.0735,
... 0.0659, 0.0923, 0.0836]
>>> newport = [0.0873, 0.0662, 0.0672, 0.0819, 0.0749, 0.0649, 0.0835,
... 0.0725]
>>> petersburg = [0.0974, 0.1352, 0.0817, 0.1016, 0.0968, 0.1064, 0.105]
>>> magadan = [0.1033, 0.0915, 0.0781, 0.0685, 0.0677, 0.0697, 0.0764,
... 0.0689]
>>> tvarminne = [0.0703, 0.1026, 0.0956, 0.0973, 0.1039, 0.1045]
>>> stats.f_oneway(tillamook, newport, petersburg, magadan, tvarminne)
(7.1210194716424473, 0.00028122423145345439)
"""
args = [np.asarray(arg, dtype=float) for arg in args]
# ANOVA on N groups, each in its own array
num_groups = len(args)
alldata = np.concatenate(args)
bign = len(alldata)
# Determine the mean of the data, and subtract that from all inputs to a
# variance (via sum_of_sq / sq_of_sum) calculation. Variance is invariance
# to a shift in location, and centering all data around zero vastly
# improves numerical stability.
offset = alldata.mean()
alldata -= offset
sstot = _sum_of_squares(alldata) - (_square_of_sums(alldata) / float(bign))
ssbn = 0
for a in args:
ssbn += _square_of_sums(a - offset) / float(len(a))
# Naming: variables ending in bn/b are for "between treatments", wn/w are
# for "within treatments"
ssbn -= (_square_of_sums(alldata) / float(bign))
sswn = sstot - ssbn
dfbn = num_groups - 1
dfwn = bign - num_groups
msb = ssbn / float(dfbn)
msw = sswn / float(dfwn)
f = msb / msw
prob = special.fdtrc(dfbn, dfwn, f) # equivalent to stats.f.sf
return F_onewayResult(f, prob)
def pearsonr(x, y):
r"""
Calculate a Pearson correlation coefficient and the p-value for testing
non-correlation.
The Pearson correlation coefficient measures the linear relationship
between two datasets. Strictly speaking, Pearson's correlation requires
that each dataset be normally distributed, and not necessarily zero-mean.
Like other correlation coefficients, this one varies between -1 and +1
with 0 implying no correlation. Correlations of -1 or +1 imply an exact
linear relationship. Positive correlations imply that as x increases, so
does y. Negative correlations imply that as x increases, y decreases.
The p-value roughly indicates the probability of an uncorrelated system
producing datasets that have a Pearson correlation at least as extreme
as the one computed from these datasets. The p-values are not entirely
reliable but are probably reasonable for datasets larger than 500 or so.
Parameters
----------
x : (N,) array_like
Input
y : (N,) array_like
Input
Returns
-------
r : float
Pearson's correlation coefficient
p-value : float
2-tailed p-value
Notes
-----
The correlation coefficient is calculated as follows:
.. math::
r_{pb} = \frac{\sum (x - m_x) (y - m_y)
}{\sqrt{\sum (x - m_x)^2 (y - m_y)^2}}
where :math:`m_x` is the mean of the vector :math:`x` and :math:`m_y` is
the mean of the vector :math:`y`.
References
----------
http://www.statsoft.com/textbook/glosp.html#Pearson%20Correlation
Examples
--------
>>> from scipy import stats
>>> a = np.array([0, 0, 0, 1, 1, 1, 1])
>>> b = np.arange(7)
>>> stats.pearsonr(a, b)
(0.8660254037844386, 0.011724811003954654)
>>> stats.pearsonr([1,2,3,4,5], [5,6,7,8,7])
(0.83205029433784372, 0.080509573298498519)
"""
# x and y should have same length.
x = np.asarray(x)
y = np.asarray(y)
n = len(x)
mx = x.mean()
my = y.mean()
xm, ym = x - mx, y - my
r_num = np.add.reduce(xm * ym)
r_den = np.sqrt(_sum_of_squares(xm) * _sum_of_squares(ym))
r = r_num / r_den
# Presumably, if abs(r) > 1, then it is only some small artifact of
# floating point arithmetic.
r = max(min(r, 1.0), -1.0)
df = n - 2
if abs(r) == 1.0:
prob = 0.0
else:
t_squared = r**2 * (df / ((1.0 - r) * (1.0 + r)))
prob = _betai(0.5*df, 0.5, df/(df+t_squared))
return r, prob
def fisher_exact(table, alternative='two-sided'):
"""Performs a Fisher exact test on a 2x2 contingency table.
Parameters
----------
table : array_like of ints
A 2x2 contingency table. Elements should be non-negative integers.
alternative : {'two-sided', 'less', 'greater'}, optional
Which alternative hypothesis to the null hypothesis the test uses.
Default is 'two-sided'.
Returns
-------
oddsratio : float
This is prior odds ratio and not a posterior estimate.
p_value : float
P-value, the probability of obtaining a distribution at least as
extreme as the one that was actually observed, assuming that the
null hypothesis is true.
See Also
--------
chi2_contingency : Chi-square test of independence of variables in a
contingency table.
Notes
-----
The calculated odds ratio is different from the one R uses. This scipy
implementation returns the (more common) "unconditional Maximum
Likelihood Estimate", while R uses the "conditional Maximum Likelihood
Estimate".
For tables with large numbers, the (inexact) chi-square test implemented
in the function `chi2_contingency` can also be used.
Examples
--------
Say we spend a few days counting whales and sharks in the Atlantic and
Indian oceans. In the Atlantic ocean we find 8 whales and 1 shark, in the
Indian ocean 2 whales and 5 sharks. Then our contingency table is::
Atlantic Indian
whales 8 2
sharks 1 5
We use this table to find the p-value:
>>> import scipy.stats as stats
>>> oddsratio, pvalue = stats.fisher_exact([[8, 2], [1, 5]])
>>> pvalue
0.0349...
The probability that we would observe this or an even more imbalanced ratio
by chance is about 3.5%. A commonly used significance level is 5%--if we
adopt that, we can therefore conclude that our observed imbalance is
statistically significant; whales prefer the Atlantic while sharks prefer
the Indian ocean.
"""
hypergeom = distributions.hypergeom
c = np.asarray(table, dtype=np.int64) # int32 is not enough for the algorithm
if not c.shape == (2, 2):
raise ValueError("The input `table` must be of shape (2, 2).")
if np.any(c < 0):
raise ValueError("All values in `table` must be nonnegative.")
if 0 in c.sum(axis=0) or 0 in c.sum(axis=1):
# If both values in a row or column are zero, the p-value is 1 and
# the odds ratio is NaN.
return np.nan, 1.0
if c[1, 0] > 0 and c[0, 1] > 0:
oddsratio = c[0, 0] * c[1, 1] / float(c[1, 0] * c[0, 1])
else:
oddsratio = np.inf
n1 = c[0, 0] + c[0, 1]
n2 = c[1, 0] + c[1, 1]
n = c[0, 0] + c[1, 0]
def binary_search(n, n1, n2, side):
"""Binary search for where to begin lower/upper halves in two-sided
test.
"""
if side == "upper":
minval = mode
maxval = n
else:
minval = 0
maxval = mode
guess = -1
while maxval - minval > 1:
if maxval == minval + 1 and guess == minval:
guess = maxval
else:
guess = (maxval + minval) // 2
pguess = hypergeom.pmf(guess, n1 + n2, n1, n)
if side == "upper":
ng = guess - 1
else:
ng = guess + 1
if pguess <= pexact < hypergeom.pmf(ng, n1 + n2, n1, n):
break
elif pguess < pexact:
maxval = guess
else:
minval = guess
if guess == -1:
guess = minval
if side == "upper":
while guess > 0 and hypergeom.pmf(guess, n1 + n2, n1, n) < pexact * epsilon:
guess -= 1
while hypergeom.pmf(guess, n1 + n2, n1, n) > pexact / epsilon:
guess += 1
else:
while hypergeom.pmf(guess, n1 + n2, n1, n) < pexact * epsilon:
guess += 1
while guess > 0 and hypergeom.pmf(guess, n1 + n2, n1, n) > pexact / epsilon:
guess -= 1
return guess
if alternative == 'less':
pvalue = hypergeom.cdf(c[0, 0], n1 + n2, n1, n)
elif alternative == 'greater':
# Same formula as the 'less' case, but with the second column.
pvalue = hypergeom.cdf(c[0, 1], n1 + n2, n1, c[0, 1] + c[1, 1])
elif alternative == 'two-sided':
mode = int(float((n + 1) * (n1 + 1)) / (n1 + n2 + 2))
pexact = hypergeom.pmf(c[0, 0], n1 + n2, n1, n)
pmode = hypergeom.pmf(mode, n1 + n2, n1, n)
epsilon = 1 - 1e-4
if np.abs(pexact - pmode) / np.maximum(pexact, pmode) <= 1 - epsilon:
return oddsratio, 1.
elif c[0, 0] < mode:
plower = hypergeom.cdf(c[0, 0], n1 + n2, n1, n)
if hypergeom.pmf(n, n1 + n2, n1, n) > pexact / epsilon:
return oddsratio, plower
guess = binary_search(n, n1, n2, "upper")
pvalue = plower + hypergeom.sf(guess - 1, n1 + n2, n1, n)
else:
pupper = hypergeom.sf(c[0, 0] - 1, n1 + n2, n1, n)
if hypergeom.pmf(0, n1 + n2, n1, n) > pexact / epsilon:
return oddsratio, pupper
guess = binary_search(n, n1, n2, "lower")
pvalue = pupper + hypergeom.cdf(guess, n1 + n2, n1, n)
else:
msg = "`alternative` should be one of {'two-sided', 'less', 'greater'}"
raise ValueError(msg)
if pvalue > 1.0:
pvalue = 1.0
return oddsratio, pvalue
SpearmanrResult = namedtuple('SpearmanrResult', ('correlation', 'pvalue'))
def spearmanr(a, b=None, axis=0, nan_policy='propagate'):
"""
Calculate a Spearman rank-order correlation coefficient and the p-value
to test for non-correlation.
The Spearman correlation is a nonparametric measure of the monotonicity
of the relationship between two datasets. Unlike the Pearson correlation,
the Spearman correlation does not assume that both datasets are normally
distributed. Like other correlation coefficients, this one varies
between -1 and +1 with 0 implying no correlation. Correlations of -1 or
+1 imply an exact monotonic relationship. Positive correlations imply that
as x increases, so does y. Negative correlations imply that as x
increases, y decreases.
The p-value roughly indicates the probability of an uncorrelated system
producing datasets that have a Spearman correlation at least as extreme
as the one computed from these datasets. The p-values are not entirely
reliable but are probably reasonable for datasets larger than 500 or so.
Parameters
----------
a, b : 1D or 2D array_like, b is optional
One or two 1-D or 2-D arrays containing multiple variables and
observations. When these are 1-D, each represents a vector of
observations of a single variable. For the behavior in the 2-D case,
see under ``axis``, below.
Both arrays need to have the same length in the ``axis`` dimension.
axis : int or None, optional
If axis=0 (default), then each column represents a variable, with
observations in the rows. If axis=1, the relationship is transposed:
each row represents a variable, while the columns contain observations.
If axis=None, then both arrays will be raveled.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
correlation : float or ndarray (2-D square)
Spearman correlation matrix or correlation coefficient (if only 2
variables are given as parameters. Correlation matrix is square with
length equal to total number of variables (columns or rows) in a and b
combined.
pvalue : float
The two-sided p-value for a hypothesis test whose null hypothesis is
that two sets of data are uncorrelated, has same dimension as rho.
Notes
-----
Changes in scipy 0.8.0: rewrite to add tie-handling, and axis.
References
----------
.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
Section 14.7
Examples
--------
>>> from scipy import stats
>>> stats.spearmanr([1,2,3,4,5], [5,6,7,8,7])
(0.82078268166812329, 0.088587005313543798)
>>> np.random.seed(1234321)
>>> x2n = np.random.randn(100, 2)
>>> y2n = np.random.randn(100, 2)
>>> stats.spearmanr(x2n)
(0.059969996999699973, 0.55338590803773591)
>>> stats.spearmanr(x2n[:,0], x2n[:,1])
(0.059969996999699973, 0.55338590803773591)
>>> rho, pval = stats.spearmanr(x2n, y2n)
>>> rho
array([[ 1. , 0.05997 , 0.18569457, 0.06258626],
[ 0.05997 , 1. , 0.110003 , 0.02534653],
[ 0.18569457, 0.110003 , 1. , 0.03488749],
[ 0.06258626, 0.02534653, 0.03488749, 1. ]])
>>> pval
array([[ 0. , 0.55338591, 0.06435364, 0.53617935],
[ 0.55338591, 0. , 0.27592895, 0.80234077],
[ 0.06435364, 0.27592895, 0. , 0.73039992],
[ 0.53617935, 0.80234077, 0.73039992, 0. ]])
>>> rho, pval = stats.spearmanr(x2n.T, y2n.T, axis=1)
>>> rho
array([[ 1. , 0.05997 , 0.18569457, 0.06258626],
[ 0.05997 , 1. , 0.110003 , 0.02534653],
[ 0.18569457, 0.110003 , 1. , 0.03488749],
[ 0.06258626, 0.02534653, 0.03488749, 1. ]])
>>> stats.spearmanr(x2n, y2n, axis=None)
(0.10816770419260482, 0.1273562188027364)
>>> stats.spearmanr(x2n.ravel(), y2n.ravel())
(0.10816770419260482, 0.1273562188027364)
>>> xint = np.random.randint(10, size=(100, 2))
>>> stats.spearmanr(xint)
(0.052760927029710199, 0.60213045837062351)
"""
a, axisout = _chk_asarray(a, axis)
a_contains_nan, nan_policy = _contains_nan(a, nan_policy)
if a_contains_nan:
a = ma.masked_invalid(a)
if a.size <= 1:
return SpearmanrResult(np.nan, np.nan)
ar = np.apply_along_axis(rankdata, axisout, a)
br = None
if b is not None:
b, axisout = _chk_asarray(b, axis)
b_contains_nan, nan_policy = _contains_nan(b, nan_policy)
if a_contains_nan or b_contains_nan:
b = ma.masked_invalid(b)
if nan_policy == 'propagate':
rho, pval = mstats_basic.spearmanr(a, b, use_ties=True)
return SpearmanrResult(rho * np.nan, pval * np.nan)
if nan_policy == 'omit':
return mstats_basic.spearmanr(a, b, use_ties=True)
br = np.apply_along_axis(rankdata, axisout, b)
n = a.shape[axisout]
rs = np.corrcoef(ar, br, rowvar=axisout)
olderr = np.seterr(divide='ignore') # rs can have elements equal to 1
try:
# clip the small negative values possibly caused by rounding
# errors before taking the square root
t = rs * np.sqrt(((n-2)/((rs+1.0)*(1.0-rs))).clip(0))
finally:
np.seterr(**olderr)
prob = 2 * distributions.t.sf(np.abs(t), n-2)
if rs.shape == (2, 2):
return SpearmanrResult(rs[1, 0], prob[1, 0])
else:
return SpearmanrResult(rs, prob)
PointbiserialrResult = namedtuple('PointbiserialrResult',
('correlation', 'pvalue'))
def pointbiserialr(x, y):
r"""
Calculate a point biserial correlation coefficient and its p-value.
The point biserial correlation is used to measure the relationship
between a binary variable, x, and a continuous variable, y. Like other
correlation coefficients, this one varies between -1 and +1 with 0
implying no correlation. Correlations of -1 or +1 imply a determinative
relationship.
This function uses a shortcut formula but produces the same result as
`pearsonr`.
Parameters
----------
x : array_like of bools
Input array.
y : array_like
Input array.
Returns
-------
correlation : float
R value
pvalue : float
2-tailed p-value
Notes
-----
`pointbiserialr` uses a t-test with ``n-1`` degrees of freedom.
It is equivalent to `pearsonr.`
The value of the point-biserial correlation can be calculated from:
.. math::
r_{pb} = \frac{\overline{Y_{1}} -
\overline{Y_{0}}}{s_{y}}\sqrt{\frac{N_{1} N_{2}}{N (N - 1))}}
Where :math:`Y_{0}` and :math:`Y_{1}` are means of the metric
observations coded 0 and 1 respectively; :math:`N_{0}` and :math:`N_{1}`
are number of observations coded 0 and 1 respectively; :math:`N` is the
total number of observations and :math:`s_{y}` is the standard
deviation of all the metric observations.
A value of :math:`r_{pb}` that is significantly different from zero is
completely equivalent to a significant difference in means between the two
groups. Thus, an independent groups t Test with :math:`N-2` degrees of
freedom may be used to test whether :math:`r_{pb}` is nonzero. The
relation between the t-statistic for comparing two independent groups and
:math:`r_{pb}` is given by:
.. math::
t = \sqrt{N - 2}\frac{r_{pb}}{\sqrt{1 - r^{2}_{pb}}}
References
----------
.. [1] J. Lev, "The Point Biserial Coefficient of Correlation", Ann. Math.
Statist., Vol. 20, no.1, pp. 125-126, 1949.
.. [2] R.F. Tate, "Correlation Between a Discrete and a Continuous
Variable. Point-Biserial Correlation.", Ann. Math. Statist., Vol. 25,
np. 3, pp. 603-607, 1954.
.. [3] http://onlinelibrary.wiley.com/doi/10.1002/9781118445112.stat06227/full
Examples
--------
>>> from scipy import stats
>>> a = np.array([0, 0, 0, 1, 1, 1, 1])
>>> b = np.arange(7)
>>> stats.pointbiserialr(a, b)
(0.8660254037844386, 0.011724811003954652)
>>> stats.pearsonr(a, b)
(0.86602540378443871, 0.011724811003954626)
>>> np.corrcoef(a, b)
array([[ 1. , 0.8660254],
[ 0.8660254, 1. ]])
"""
rpb, prob = pearsonr(x, y)
return PointbiserialrResult(rpb, prob)
KendalltauResult = namedtuple('KendalltauResult', ('correlation', 'pvalue'))
def kendalltau(x, y, initial_lexsort=None, nan_policy='propagate'):
"""
Calculate Kendall's tau, a correlation measure for ordinal data.
Kendall's tau is a measure of the correspondence between two rankings.
Values close to 1 indicate strong agreement, values close to -1 indicate
strong disagreement. This is the 1945 "tau-b" version of Kendall's
tau [2]_, which can account for ties and which reduces to the 1938 "tau-a"
version [1]_ in absence of ties.
Parameters
----------
x, y : array_like
Arrays of rankings, of the same shape. If arrays are not 1-D, they will
be flattened to 1-D.
initial_lexsort : bool, optional
Unused (deprecated).
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'. Note that if the input contains nan
'omit' delegates to mstats_basic.kendalltau(), which has a different
implementation.
Returns
-------
correlation : float
The tau statistic.
pvalue : float
The two-sided p-value for a hypothesis test whose null hypothesis is
an absence of association, tau = 0.
See also
--------
spearmanr : Calculates a Spearman rank-order correlation coefficient.
theilslopes : Computes the Theil-Sen estimator for a set of points (x, y).
weightedtau : Computes a weighted version of Kendall's tau.
Notes
-----
The definition of Kendall's tau that is used is [2]_::
tau = (P - Q) / sqrt((P + Q + T) * (P + Q + U))
where P is the number of concordant pairs, Q the number of discordant
pairs, T the number of ties only in `x`, and U the number of ties only in
`y`. If a tie occurs for the same pair in both `x` and `y`, it is not
added to either T or U.
References
----------
.. [1] Maurice G. Kendall, "A New Measure of Rank Correlation", Biometrika
Vol. 30, No. 1/2, pp. 81-93, 1938.
.. [2] Maurice G. Kendall, "The treatment of ties in ranking problems",
Biometrika Vol. 33, No. 3, pp. 239-251. 1945.
.. [3] Gottfried E. Noether, "Elements of Nonparametric Statistics", John
Wiley & Sons, 1967.
.. [4] Peter M. Fenwick, "A new data structure for cumulative frequency
tables", Software: Practice and Experience, Vol. 24, No. 3,
pp. 327-336, 1994.
Examples
--------
>>> from scipy import stats
>>> x1 = [12, 2, 1, 12, 2]
>>> x2 = [1, 4, 7, 1, 0]
>>> tau, p_value = stats.kendalltau(x1, x2)
>>> tau
-0.47140452079103173
>>> p_value
0.2827454599327748
"""
x = np.asarray(x).ravel()
y = np.asarray(y).ravel()
if x.size != y.size:
raise ValueError("All inputs to `kendalltau` must be of the same size, "
"found x-size %s and y-size %s" % (x.size, y.size))
elif not x.size or not y.size:
return KendalltauResult(np.nan, np.nan) # Return NaN if arrays are empty
# check both x and y
cnx, npx = _contains_nan(x, nan_policy)
cny, npy = _contains_nan(y, nan_policy)
contains_nan = cnx or cny
if npx == 'omit' or npy == 'omit':
nan_policy = 'omit'
if contains_nan and nan_policy == 'propagate':
return KendalltauResult(np.nan, np.nan)
elif contains_nan and nan_policy == 'omit':
x = ma.masked_invalid(x)
y = ma.masked_invalid(y)
return mstats_basic.kendalltau(x, y)
if initial_lexsort is not None: # deprecate to drop!
warnings.warn('"initial_lexsort" is gone!')
def count_rank_tie(ranks):
cnt = np.bincount(ranks).astype('int64', copy=False)
cnt = cnt[cnt > 1]
return ((cnt * (cnt - 1) // 2).sum(),
(cnt * (cnt - 1.) * (cnt - 2)).sum(),
(cnt * (cnt - 1.) * (2*cnt + 5)).sum())
size = x.size
perm = np.argsort(y) # sort on y and convert y to dense ranks
x, y = x[perm], y[perm]
y = np.r_[True, y[1:] != y[:-1]].cumsum(dtype=np.intp)
# stable sort on x and convert x to dense ranks
perm = np.argsort(x, kind='mergesort')
x, y = x[perm], y[perm]
x = np.r_[True, x[1:] != x[:-1]].cumsum(dtype=np.intp)
dis = _kendall_dis(x, y) # discordant pairs
obs = np.r_[True, (x[1:] != x[:-1]) | (y[1:] != y[:-1]), True]
cnt = np.diff(np.where(obs)[0]).astype('int64', copy=False)
ntie = (cnt * (cnt - 1) // 2).sum() # joint ties
xtie, x0, x1 = count_rank_tie(x) # ties in x, stats
ytie, y0, y1 = count_rank_tie(y) # ties in y, stats
tot = (size * (size - 1)) // 2
if xtie == tot or ytie == tot:
return KendalltauResult(np.nan, np.nan)
# Note that tot = con + dis + (xtie - ntie) + (ytie - ntie) + ntie
# = con + dis + xtie + ytie - ntie
con_minus_dis = tot - xtie - ytie + ntie - 2 * dis
tau = con_minus_dis / np.sqrt(tot - xtie) / np.sqrt(tot - ytie)
# Limit range to fix computational errors
tau = min(1., max(-1., tau))
# con_minus_dis is approx normally distributed with this variance [3]_
var = (size * (size - 1) * (2.*size + 5) - x1 - y1) / 18. + (
2. * xtie * ytie) / (size * (size - 1)) + x0 * y0 / (9. *
size * (size - 1) * (size - 2))
pvalue = special.erfc(np.abs(con_minus_dis) / np.sqrt(var) / np.sqrt(2))
# Limit range to fix computational errors
return KendalltauResult(min(1., max(-1., tau)), pvalue)
WeightedTauResult = namedtuple('WeightedTauResult', ('correlation', 'pvalue'))
def weightedtau(x, y, rank=True, weigher=None, additive=True):
r"""
Compute a weighted version of Kendall's :math:`\tau`.
The weighted :math:`\tau` is a weighted version of Kendall's
:math:`\tau` in which exchanges of high weight are more influential than
exchanges of low weight. The default parameters compute the additive
hyperbolic version of the index, :math:`\tau_\mathrm h`, which has
been shown to provide the best balance between important and
unimportant elements [1]_.
The weighting is defined by means of a rank array, which assigns a
nonnegative rank to each element, and a weigher function, which
assigns a weight based from the rank to each element. The weight of an
exchange is then the sum or the product of the weights of the ranks of
the exchanged elements. The default parameters compute
:math:`\tau_\mathrm h`: an exchange between elements with rank
:math:`r` and :math:`s` (starting from zero) has weight
:math:`1/(r+1) + 1/(s+1)`.
Specifying a rank array is meaningful only if you have in mind an
external criterion of importance. If, as it usually happens, you do
not have in mind a specific rank, the weighted :math:`\tau` is
defined by averaging the values obtained using the decreasing
lexicographical rank by (`x`, `y`) and by (`y`, `x`). This is the
behavior with default parameters.
Note that if you are computing the weighted :math:`\tau` on arrays of
ranks, rather than of scores (i.e., a larger value implies a lower
rank) you must negate the ranks, so that elements of higher rank are
associated with a larger value.
Parameters
----------
x, y : array_like
Arrays of scores, of the same shape. If arrays are not 1-D, they will
be flattened to 1-D.
rank: array_like of ints or bool, optional
A nonnegative rank assigned to each element. If it is None, the
decreasing lexicographical rank by (`x`, `y`) will be used: elements of
higher rank will be those with larger `x`-values, using `y`-values to
break ties (in particular, swapping `x` and `y` will give a different
result). If it is False, the element indices will be used
directly as ranks. The default is True, in which case this
function returns the average of the values obtained using the
decreasing lexicographical rank by (`x`, `y`) and by (`y`, `x`).
weigher : callable, optional
The weigher function. Must map nonnegative integers (zero
representing the most important element) to a nonnegative weight.
The default, None, provides hyperbolic weighing, that is,
rank :math:`r` is mapped to weight :math:`1/(r+1)`.
additive : bool, optional
If True, the weight of an exchange is computed by adding the
weights of the ranks of the exchanged elements; otherwise, the weights
are multiplied. The default is True.
Returns
-------
correlation : float
The weighted :math:`\tau` correlation index.
pvalue : float
Presently ``np.nan``, as the null statistics is unknown (even in the
additive hyperbolic case).
See also
--------
kendalltau : Calculates Kendall's tau.
spearmanr : Calculates a Spearman rank-order correlation coefficient.
theilslopes : Computes the Theil-Sen estimator for a set of points (x, y).
Notes
-----
This function uses an :math:`O(n \log n)`, mergesort-based algorithm
[1]_ that is a weighted extension of Knight's algorithm for Kendall's
:math:`\tau` [2]_. It can compute Shieh's weighted :math:`\tau` [3]_
between rankings without ties (i.e., permutations) by setting
`additive` and `rank` to False, as the definition given in [1]_ is a
generalization of Shieh's.
NaNs are considered the smallest possible score.
.. versionadded:: 0.19.0
References
----------
.. [1] Sebastiano Vigna, "A weighted correlation index for rankings with
ties", Proceedings of the 24th international conference on World
Wide Web, pp. 1166-1176, ACM, 2015.
.. [2] W.R. Knight, "A Computer Method for Calculating Kendall's Tau with
Ungrouped Data", Journal of the American Statistical Association,
Vol. 61, No. 314, Part 1, pp. 436-439, 1966.
.. [3] Grace S. Shieh. "A weighted Kendall's tau statistic", Statistics &
Probability Letters, Vol. 39, No. 1, pp. 17-24, 1998.
Examples
--------
>>> from scipy import stats
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> tau, p_value = stats.weightedtau(x, y)
>>> tau
-0.56694968153682723
>>> p_value
nan
>>> tau, p_value = stats.weightedtau(x, y, additive=False)
>>> tau
-0.62205716951801038
NaNs are considered the smallest possible score:
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, np.nan]
>>> tau, _ = stats.weightedtau(x, y)
>>> tau
-0.56694968153682723
This is exactly Kendall's tau:
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> tau, _ = stats.weightedtau(x, y, weigher=lambda x: 1)
>>> tau
-0.47140452079103173
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> stats.weightedtau(x, y, rank=None)
WeightedTauResult(correlation=-0.4157652301037516, pvalue=nan)
>>> stats.weightedtau(y, x, rank=None)
WeightedTauResult(correlation=-0.7181341329699028, pvalue=nan)
"""
x = np.asarray(x).ravel()
y = np.asarray(y).ravel()
if x.size != y.size:
raise ValueError("All inputs to `weightedtau` must be of the same size, "
"found x-size %s and y-size %s" % (x.size, y.size))
if not x.size:
return WeightedTauResult(np.nan, np.nan) # Return NaN if arrays are empty
# If there are NaNs we apply _toint64()
if np.isnan(np.sum(x)):
x = _toint64(x)
if np.isnan(np.sum(x)):
y = _toint64(y)
# Reduce to ranks unsupported types
if x.dtype != y.dtype:
if x.dtype != np.int64:
x = _toint64(x)
if y.dtype != np.int64:
y = _toint64(y)
else:
if x.dtype not in (np.int32, np.int64, np.float32, np.float64):
x = _toint64(x)
y = _toint64(y)
if rank is True:
return WeightedTauResult((
_weightedrankedtau(x, y, None, weigher, additive) +
_weightedrankedtau(y, x, None, weigher, additive)
) / 2, np.nan)
if rank is False:
rank = np.arange(x.size, dtype=np.intp)
elif rank is not None:
rank = np.asarray(rank).ravel()
if rank.size != x.size:
raise ValueError("All inputs to `weightedtau` must be of the same size, "
"found x-size %s and rank-size %s" % (x.size, rank.size))
return WeightedTauResult(_weightedrankedtau(x, y, rank, weigher, additive), np.nan)
#####################################
# INFERENTIAL STATISTICS #
#####################################
Ttest_1sampResult = namedtuple('Ttest_1sampResult', ('statistic', 'pvalue'))
def ttest_1samp(a, popmean, axis=0, nan_policy='propagate'):
"""
Calculate the T-test for the mean of ONE group of scores.
This is a two-sided test for the null hypothesis that the expected value
(mean) of a sample of independent observations `a` is equal to the given
population mean, `popmean`.
Parameters
----------
a : array_like
sample observation
popmean : float or array_like
expected value in null hypothesis. If array_like, then it must have the
same shape as `a` excluding the axis dimension
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole
array `a`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float or array
t-statistic
pvalue : float or array
two-tailed p-value
Examples
--------
>>> from scipy import stats
>>> np.random.seed(7654567) # fix seed to get the same result
>>> rvs = stats.norm.rvs(loc=5, scale=10, size=(50,2))
Test if mean of random sample is equal to true mean, and different mean.
We reject the null hypothesis in the second case and don't reject it in
the first case.
>>> stats.ttest_1samp(rvs,5.0)
(array([-0.68014479, -0.04323899]), array([ 0.49961383, 0.96568674]))
>>> stats.ttest_1samp(rvs,0.0)
(array([ 2.77025808, 4.11038784]), array([ 0.00789095, 0.00014999]))
Examples using axis and non-scalar dimension for population mean.
>>> stats.ttest_1samp(rvs,[5.0,0.0])
(array([-0.68014479, 4.11038784]), array([ 4.99613833e-01, 1.49986458e-04]))
>>> stats.ttest_1samp(rvs.T,[5.0,0.0],axis=1)
(array([-0.68014479, 4.11038784]), array([ 4.99613833e-01, 1.49986458e-04]))
>>> stats.ttest_1samp(rvs,[[5.0],[0.0]])
(array([[-0.68014479, -0.04323899],
[ 2.77025808, 4.11038784]]), array([[ 4.99613833e-01, 9.65686743e-01],
[ 7.89094663e-03, 1.49986458e-04]]))
"""
a, axis = _chk_asarray(a, axis)
contains_nan, nan_policy = _contains_nan(a, nan_policy)
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
return mstats_basic.ttest_1samp(a, popmean, axis)
n = a.shape[axis]
df = n - 1
d = np.mean(a, axis) - popmean
v = np.var(a, axis, ddof=1)
denom = np.sqrt(v / float(n))
with np.errstate(divide='ignore', invalid='ignore'):
t = np.divide(d, denom)
t, prob = _ttest_finish(df, t)
return Ttest_1sampResult(t, prob)
def _ttest_finish(df, t):
"""Common code between all 3 t-test functions."""
prob = distributions.t.sf(np.abs(t), df) * 2 # use np.abs to get upper tail
if t.ndim == 0:
t = t[()]
return t, prob
def _ttest_ind_from_stats(mean1, mean2, denom, df):
d = mean1 - mean2
with np.errstate(divide='ignore', invalid='ignore'):
t = np.divide(d, denom)
t, prob = _ttest_finish(df, t)
return (t, prob)
def _unequal_var_ttest_denom(v1, n1, v2, n2):
vn1 = v1 / n1
vn2 = v2 / n2
with np.errstate(divide='ignore', invalid='ignore'):
df = (vn1 + vn2)**2 / (vn1**2 / (n1 - 1) + vn2**2 / (n2 - 1))
# If df is undefined, variances are zero (assumes n1 > 0 & n2 > 0).
# Hence it doesn't matter what df is as long as it's not NaN.
df = np.where(np.isnan(df), 1, df)
denom = np.sqrt(vn1 + vn2)
return df, denom
def _equal_var_ttest_denom(v1, n1, v2, n2):
df = n1 + n2 - 2.0
svar = ((n1 - 1) * v1 + (n2 - 1) * v2) / df
denom = np.sqrt(svar * (1.0 / n1 + 1.0 / n2))
return df, denom
Ttest_indResult = namedtuple('Ttest_indResult', ('statistic', 'pvalue'))
def ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2,
equal_var=True):
"""
T-test for means of two independent samples from descriptive statistics.
This is a two-sided test for the null hypothesis that two independent
samples have identical average (expected) values.
Parameters
----------
mean1 : array_like
The mean(s) of sample 1.
std1 : array_like
The standard deviation(s) of sample 1.
nobs1 : array_like
The number(s) of observations of sample 1.
mean2 : array_like
The mean(s) of sample 2
std2 : array_like
The standard deviations(s) of sample 2.
nobs2 : array_like
The number(s) of observations of sample 2.
equal_var : bool, optional
If True (default), perform a standard independent 2 sample test
that assumes equal population variances [1]_.
If False, perform Welch's t-test, which does not assume equal
population variance [2]_.
Returns
-------
statistic : float or array
The calculated t-statistics
pvalue : float or array
The two-tailed p-value.
See Also
--------
scipy.stats.ttest_ind
Notes
-----
.. versionadded:: 0.16.0
References
----------
.. [1] http://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test
.. [2] http://en.wikipedia.org/wiki/Welch%27s_t_test
Examples
--------
Suppose we have the summary data for two samples, as follows::
Sample Sample
Size Mean Variance
Sample 1 13 15.0 87.5
Sample 2 11 12.0 39.0
Apply the t-test to this data (with the assumption that the population
variances are equal):
>>> from scipy.stats import ttest_ind_from_stats
>>> ttest_ind_from_stats(mean1=15.0, std1=np.sqrt(87.5), nobs1=13,
... mean2=12.0, std2=np.sqrt(39.0), nobs2=11)
Ttest_indResult(statistic=0.9051358093310269, pvalue=0.3751996797581487)
For comparison, here is the data from which those summary statistics
were taken. With this data, we can compute the same result using
`scipy.stats.ttest_ind`:
>>> a = np.array([1, 3, 4, 6, 11, 13, 15, 19, 22, 24, 25, 26, 26])
>>> b = np.array([2, 4, 6, 9, 11, 13, 14, 15, 18, 19, 21])
>>> from scipy.stats import ttest_ind
>>> ttest_ind(a, b)
Ttest_indResult(statistic=0.905135809331027, pvalue=0.3751996797581486)
"""
if equal_var:
df, denom = _equal_var_ttest_denom(std1**2, nobs1, std2**2, nobs2)
else:
df, denom = _unequal_var_ttest_denom(std1**2, nobs1,
std2**2, nobs2)
res = _ttest_ind_from_stats(mean1, mean2, denom, df)
return Ttest_indResult(*res)
def ttest_ind(a, b, axis=0, equal_var=True, nan_policy='propagate'):
"""
Calculate the T-test for the means of *two independent* samples of scores.
This is a two-sided test for the null hypothesis that 2 independent samples
have identical average (expected) values. This test assumes that the
populations have identical variances by default.
Parameters
----------
a, b : array_like
The arrays must have the same shape, except in the dimension
corresponding to `axis` (the first, by default).
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole
arrays, `a`, and `b`.
equal_var : bool, optional
If True (default), perform a standard independent 2 sample test
that assumes equal population variances [1]_.
If False, perform Welch's t-test, which does not assume equal
population variance [2]_.
.. versionadded:: 0.11.0
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float or array
The calculated t-statistic.
pvalue : float or array
The two-tailed p-value.
Notes
-----
We can use this test, if we observe two independent samples from
the same or different population, e.g. exam scores of boys and
girls or of two ethnic groups. The test measures whether the
average (expected) value differs significantly across samples. If
we observe a large p-value, for example larger than 0.05 or 0.1,
then we cannot reject the null hypothesis of identical average scores.
If the p-value is smaller than the threshold, e.g. 1%, 5% or 10%,
then we reject the null hypothesis of equal averages.
References
----------
.. [1] http://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test
.. [2] http://en.wikipedia.org/wiki/Welch%27s_t_test
Examples
--------
>>> from scipy import stats
>>> np.random.seed(12345678)
Test with sample with identical means:
>>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500)
>>> rvs2 = stats.norm.rvs(loc=5,scale=10,size=500)
>>> stats.ttest_ind(rvs1,rvs2)
(0.26833823296239279, 0.78849443369564776)
>>> stats.ttest_ind(rvs1,rvs2, equal_var = False)
(0.26833823296239279, 0.78849452749500748)
`ttest_ind` underestimates p for unequal variances:
>>> rvs3 = stats.norm.rvs(loc=5, scale=20, size=500)
>>> stats.ttest_ind(rvs1, rvs3)
(-0.46580283298287162, 0.64145827413436174)
>>> stats.ttest_ind(rvs1, rvs3, equal_var = False)
(-0.46580283298287162, 0.64149646246569292)
When n1 != n2, the equal variance t-statistic is no longer equal to the
unequal variance t-statistic:
>>> rvs4 = stats.norm.rvs(loc=5, scale=20, size=100)
>>> stats.ttest_ind(rvs1, rvs4)
(-0.99882539442782481, 0.3182832709103896)
>>> stats.ttest_ind(rvs1, rvs4, equal_var = False)
(-0.69712570584654099, 0.48716927725402048)
T-test with different means, variance, and n:
>>> rvs5 = stats.norm.rvs(loc=8, scale=20, size=100)
>>> stats.ttest_ind(rvs1, rvs5)
(-1.4679669854490653, 0.14263895620529152)
>>> stats.ttest_ind(rvs1, rvs5, equal_var = False)
(-0.94365973617132992, 0.34744170334794122)
"""
a, b, axis = _chk2_asarray(a, b, axis)
# check both a and b
cna, npa = _contains_nan(a, nan_policy)
cnb, npb = _contains_nan(b, nan_policy)
contains_nan = cna or cnb
if npa == 'omit' or npb == 'omit':
nan_policy = 'omit'
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
b = ma.masked_invalid(b)
return mstats_basic.ttest_ind(a, b, axis, equal_var)
if a.size == 0 or b.size == 0:
return Ttest_indResult(np.nan, np.nan)
v1 = np.var(a, axis, ddof=1)
v2 = np.var(b, axis, ddof=1)
n1 = a.shape[axis]
n2 = b.shape[axis]
if equal_var:
df, denom = _equal_var_ttest_denom(v1, n1, v2, n2)
else:
df, denom = _unequal_var_ttest_denom(v1, n1, v2, n2)
res = _ttest_ind_from_stats(np.mean(a, axis), np.mean(b, axis), denom, df)
return Ttest_indResult(*res)
Ttest_relResult = namedtuple('Ttest_relResult', ('statistic', 'pvalue'))
def ttest_rel(a, b, axis=0, nan_policy='propagate'):
"""
Calculate the T-test on TWO RELATED samples of scores, a and b.
This is a two-sided test for the null hypothesis that 2 related or
repeated samples have identical average (expected) values.
Parameters
----------
a, b : array_like
The arrays must have the same shape.
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole
arrays, `a`, and `b`.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float or array
t-statistic
pvalue : float or array
two-tailed p-value
Notes
-----
Examples for the use are scores of the same set of student in
different exams, or repeated sampling from the same units. The
test measures whether the average score differs significantly
across samples (e.g. exams). If we observe a large p-value, for
example greater than 0.05 or 0.1 then we cannot reject the null
hypothesis of identical average scores. If the p-value is smaller
than the threshold, e.g. 1%, 5% or 10%, then we reject the null
hypothesis of equal averages. Small p-values are associated with
large t-statistics.
References
----------
https://en.wikipedia.org/wiki/T-test#Dependent_t-test_for_paired_samples
Examples
--------
>>> from scipy import stats
>>> np.random.seed(12345678) # fix random seed to get same numbers
>>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500)
>>> rvs2 = (stats.norm.rvs(loc=5,scale=10,size=500) +
... stats.norm.rvs(scale=0.2,size=500))
>>> stats.ttest_rel(rvs1,rvs2)
(0.24101764965300962, 0.80964043445811562)
>>> rvs3 = (stats.norm.rvs(loc=8,scale=10,size=500) +
... stats.norm.rvs(scale=0.2,size=500))
>>> stats.ttest_rel(rvs1,rvs3)
(-3.9995108708727933, 7.3082402191726459e-005)
"""
a, b, axis = _chk2_asarray(a, b, axis)
cna, npa = _contains_nan(a, nan_policy)
cnb, npb = _contains_nan(b, nan_policy)
contains_nan = cna or cnb
if npa == 'omit' or npb == 'omit':
nan_policy = 'omit'
if contains_nan and nan_policy == 'omit':
a = ma.masked_invalid(a)
b = ma.masked_invalid(b)
m = ma.mask_or(ma.getmask(a), ma.getmask(b))
aa = ma.array(a, mask=m, copy=True)
bb = ma.array(b, mask=m, copy=True)
return mstats_basic.ttest_rel(aa, bb, axis)
if a.shape[axis] != b.shape[axis]:
raise ValueError('unequal length arrays')
if a.size == 0 or b.size == 0:
return np.nan, np.nan
n = a.shape[axis]
df = float(n - 1)
d = (a - b).astype(np.float64)
v = np.var(d, axis, ddof=1)
dm = np.mean(d, axis)
denom = np.sqrt(v / float(n))
with np.errstate(divide='ignore', invalid='ignore'):
t = np.divide(dm, denom)
t, prob = _ttest_finish(df, t)
return Ttest_relResult(t, prob)
KstestResult = namedtuple('KstestResult', ('statistic', 'pvalue'))
def kstest(rvs, cdf, args=(), N=20, alternative='two-sided', mode='approx'):
"""
Perform the Kolmogorov-Smirnov test for goodness of fit.
This performs a test of the distribution G(x) of an observed
random variable against a given distribution F(x). Under the null
hypothesis the two distributions are identical, G(x)=F(x). The
alternative hypothesis can be either 'two-sided' (default), 'less'
or 'greater'. The KS test is only valid for continuous distributions.
Parameters
----------
rvs : str, array or callable
If a string, it should be the name of a distribution in `scipy.stats`.
If an array, it should be a 1-D array of observations of random
variables.
If a callable, it should be a function to generate random variables;
it is required to have a keyword argument `size`.
cdf : str or callable
If a string, it should be the name of a distribution in `scipy.stats`.
If `rvs` is a string then `cdf` can be False or the same as `rvs`.
If a callable, that callable is used to calculate the cdf.
args : tuple, sequence, optional
Distribution parameters, used if `rvs` or `cdf` are strings.
N : int, optional
Sample size if `rvs` is string or callable. Default is 20.
alternative : {'two-sided', 'less','greater'}, optional
Defines the alternative hypothesis (see explanation above).
Default is 'two-sided'.
mode : 'approx' (default) or 'asymp', optional
Defines the distribution used for calculating the p-value.
- 'approx' : use approximation to exact distribution of test statistic
- 'asymp' : use asymptotic distribution of test statistic
Returns
-------
statistic : float
KS test statistic, either D, D+ or D-.
pvalue : float
One-tailed or two-tailed p-value.
Notes
-----
In the one-sided test, the alternative is that the empirical
cumulative distribution function of the random variable is "less"
or "greater" than the cumulative distribution function F(x) of the
hypothesis, ``G(x)<=F(x)``, resp. ``G(x)>=F(x)``.
Examples
--------
>>> from scipy import stats
>>> x = np.linspace(-15, 15, 9)
>>> stats.kstest(x, 'norm')
(0.44435602715924361, 0.038850142705171065)
>>> np.random.seed(987654321) # set random seed to get the same result
>>> stats.kstest('norm', False, N=100)
(0.058352892479417884, 0.88531190944151261)
The above lines are equivalent to:
>>> np.random.seed(987654321)
>>> stats.kstest(stats.norm.rvs(size=100), 'norm')
(0.058352892479417884, 0.88531190944151261)
*Test against one-sided alternative hypothesis*
Shift distribution to larger values, so that ``cdf_dgp(x) < norm.cdf(x)``:
>>> np.random.seed(987654321)
>>> x = stats.norm.rvs(loc=0.2, size=100)
>>> stats.kstest(x,'norm', alternative = 'less')
(0.12464329735846891, 0.040989164077641749)
Reject equal distribution against alternative hypothesis: less
>>> stats.kstest(x,'norm', alternative = 'greater')
(0.0072115233216311081, 0.98531158590396395)
Don't reject equal distribution against alternative hypothesis: greater
>>> stats.kstest(x,'norm', mode='asymp')
(0.12464329735846891, 0.08944488871182088)
*Testing t distributed random variables against normal distribution*
With 100 degrees of freedom the t distribution looks close to the normal
distribution, and the K-S test does not reject the hypothesis that the
sample came from the normal distribution:
>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(100,size=100),'norm')
(0.072018929165471257, 0.67630062862479168)
With 3 degrees of freedom the t distribution looks sufficiently different
from the normal distribution, that we can reject the hypothesis that the
sample came from the normal distribution at the 10% level:
>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(3,size=100),'norm')
(0.131016895759829, 0.058826222555312224)
"""
if isinstance(rvs, string_types):
if (not cdf) or (cdf == rvs):
cdf = getattr(distributions, rvs).cdf
rvs = getattr(distributions, rvs).rvs
else:
raise AttributeError("if rvs is string, cdf has to be the "
"same distribution")
if isinstance(cdf, string_types):
cdf = getattr(distributions, cdf).cdf
if callable(rvs):
kwds = {'size': N}
vals = np.sort(rvs(*args, **kwds))
else:
vals = np.sort(rvs)
N = len(vals)
cdfvals = cdf(vals, *args)
# to not break compatibility with existing code
if alternative == 'two_sided':
alternative = 'two-sided'
if alternative in ['two-sided', 'greater']:
Dplus = (np.arange(1.0, N + 1)/N - cdfvals).max()
if alternative == 'greater':
return KstestResult(Dplus, distributions.ksone.sf(Dplus, N))
if alternative in ['two-sided', 'less']:
Dmin = (cdfvals - np.arange(0.0, N)/N).max()
if alternative == 'less':
return KstestResult(Dmin, distributions.ksone.sf(Dmin, N))
if alternative == 'two-sided':
D = np.max([Dplus, Dmin])
if mode == 'asymp':
return KstestResult(D, distributions.kstwobign.sf(D * np.sqrt(N)))
if mode == 'approx':
pval_two = distributions.kstwobign.sf(D * np.sqrt(N))
if N > 2666 or pval_two > 0.80 - N*0.3/1000:
return KstestResult(D, pval_two)
else:
return KstestResult(D, 2 * distributions.ksone.sf(D, N))
# Map from names to lambda_ values used in power_divergence().
_power_div_lambda_names = {
"pearson": 1,
"log-likelihood": 0,
"freeman-tukey": -0.5,
"mod-log-likelihood": -1,
"neyman": -2,
"cressie-read": 2/3,
}
def _count(a, axis=None):
"""
Count the number of non-masked elements of an array.
This function behaves like np.ma.count(), but is much faster
for ndarrays.
"""
if hasattr(a, 'count'):
num = a.count(axis=axis)
if isinstance(num, np.ndarray) and num.ndim == 0:
# In some cases, the `count` method returns a scalar array (e.g.
# np.array(3)), but we want a plain integer.
num = int(num)
else:
if axis is None:
num = a.size
else:
num = a.shape[axis]
return num
Power_divergenceResult = namedtuple('Power_divergenceResult',
('statistic', 'pvalue'))
def power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None):
"""
Cressie-Read power divergence statistic and goodness of fit test.
This function tests the null hypothesis that the categorical data
has the given frequencies, using the Cressie-Read power divergence
statistic.
Parameters
----------
f_obs : array_like
Observed frequencies in each category.
f_exp : array_like, optional
Expected frequencies in each category. By default the categories are
assumed to be equally likely.
ddof : int, optional
"Delta degrees of freedom": adjustment to the degrees of freedom
for the p-value. The p-value is computed using a chi-squared
distribution with ``k - 1 - ddof`` degrees of freedom, where `k`
is the number of observed frequencies. The default value of `ddof`
is 0.
axis : int or None, optional
The axis of the broadcast result of `f_obs` and `f_exp` along which to
apply the test. If axis is None, all values in `f_obs` are treated
as a single data set. Default is 0.
lambda_ : float or str, optional
`lambda_` gives the power in the Cressie-Read power divergence
statistic. The default is 1. For convenience, `lambda_` may be
assigned one of the following strings, in which case the
corresponding numerical value is used::
String Value Description
"pearson" 1 Pearson's chi-squared statistic.
In this case, the function is
equivalent to `stats.chisquare`.
"log-likelihood" 0 Log-likelihood ratio. Also known as
the G-test [3]_.
"freeman-tukey" -1/2 Freeman-Tukey statistic.
"mod-log-likelihood" -1 Modified log-likelihood ratio.
"neyman" -2 Neyman's statistic.
"cressie-read" 2/3 The power recommended in [5]_.
Returns
-------
statistic : float or ndarray
The Cressie-Read power divergence test statistic. The value is
a float if `axis` is None or if` `f_obs` and `f_exp` are 1-D.
pvalue : float or ndarray
The p-value of the test. The value is a float if `ddof` and the
return value `stat` are scalars.
See Also
--------
chisquare
Notes
-----
This test is invalid when the observed or expected frequencies in each
category are too small. A typical rule is that all of the observed
and expected frequencies should be at least 5.
When `lambda_` is less than zero, the formula for the statistic involves
dividing by `f_obs`, so a warning or error may be generated if any value
in `f_obs` is 0.
Similarly, a warning or error may be generated if any value in `f_exp` is
zero when `lambda_` >= 0.
The default degrees of freedom, k-1, are for the case when no parameters
of the distribution are estimated. If p parameters are estimated by
efficient maximum likelihood then the correct degrees of freedom are
k-1-p. If the parameters are estimated in a different way, then the
dof can be between k-1-p and k-1. However, it is also possible that
the asymptotic distribution is not a chisquare, in which case this
test is not appropriate.
This function handles masked arrays. If an element of `f_obs` or `f_exp`
is masked, then data at that position is ignored, and does not count
towards the size of the data set.
.. versionadded:: 0.13.0
References
----------
.. [1] Lowry, Richard. "Concepts and Applications of Inferential
Statistics". Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html
.. [2] "Chi-squared test", http://en.wikipedia.org/wiki/Chi-squared_test
.. [3] "G-test", http://en.wikipedia.org/wiki/G-test
.. [4] Sokal, R. R. and Rohlf, F. J. "Biometry: the principles and
practice of statistics in biological research", New York: Freeman
(1981)
.. [5] Cressie, N. and Read, T. R. C., "Multinomial Goodness-of-Fit
Tests", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984),
pp. 440-464.
Examples
--------
(See `chisquare` for more examples.)
When just `f_obs` is given, it is assumed that the expected frequencies
are uniform and given by the mean of the observed frequencies. Here we
perform a G-test (i.e. use the log-likelihood ratio statistic):
>>> from scipy.stats import power_divergence
>>> power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood')
(2.006573162632538, 0.84823476779463769)
The expected frequencies can be given with the `f_exp` argument:
>>> power_divergence([16, 18, 16, 14, 12, 12],
... f_exp=[16, 16, 16, 16, 16, 8],
... lambda_='log-likelihood')
(3.3281031458963746, 0.6495419288047497)
When `f_obs` is 2-D, by default the test is applied to each column.
>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
>>> obs.shape
(6, 2)
>>> power_divergence(obs, lambda_="log-likelihood")
(array([ 2.00657316, 6.77634498]), array([ 0.84823477, 0.23781225]))
By setting ``axis=None``, the test is applied to all data in the array,
which is equivalent to applying the test to the flattened array.
>>> power_divergence(obs, axis=None)
(23.31034482758621, 0.015975692534127565)
>>> power_divergence(obs.ravel())
(23.31034482758621, 0.015975692534127565)
`ddof` is the change to make to the default degrees of freedom.
>>> power_divergence([16, 18, 16, 14, 12, 12], ddof=1)
(2.0, 0.73575888234288467)
The calculation of the p-values is done by broadcasting the
test statistic with `ddof`.
>>> power_divergence([16, 18, 16, 14, 12, 12], ddof=[0,1,2])
(2.0, array([ 0.84914504, 0.73575888, 0.5724067 ]))
`f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has
shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting
`f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared
statistics, we must use ``axis=1``:
>>> power_divergence([16, 18, 16, 14, 12, 12],
... f_exp=[[16, 16, 16, 16, 16, 8],
... [8, 20, 20, 16, 12, 12]],
... axis=1)
(array([ 3.5 , 9.25]), array([ 0.62338763, 0.09949846]))
"""
# Convert the input argument `lambda_` to a numerical value.
if isinstance(lambda_, string_types):
if lambda_ not in _power_div_lambda_names:
names = repr(list(_power_div_lambda_names.keys()))[1:-1]
raise ValueError("invalid string for lambda_: {0!r}. Valid strings "
"are {1}".format(lambda_, names))
lambda_ = _power_div_lambda_names[lambda_]
elif lambda_ is None:
lambda_ = 1
f_obs = np.asanyarray(f_obs)
if f_exp is not None:
f_exp = np.atleast_1d(np.asanyarray(f_exp))
else:
# Compute the equivalent of
# f_exp = f_obs.mean(axis=axis, keepdims=True)
# Older versions of numpy do not have the 'keepdims' argument, so
# we have to do a little work to achieve the same result.
# Ignore 'invalid' errors so the edge case of a data set with length 0
# is handled without spurious warnings.
with np.errstate(invalid='ignore'):
f_exp = np.atleast_1d(f_obs.mean(axis=axis))
if axis is not None:
reduced_shape = list(f_obs.shape)
reduced_shape[axis] = 1
f_exp.shape = reduced_shape
# `terms` is the array of terms that are summed along `axis` to create
# the test statistic. We use some specialized code for a few special
# cases of lambda_.
if lambda_ == 1:
# Pearson's chi-squared statistic
terms = (f_obs - f_exp)**2 / f_exp
elif lambda_ == 0:
# Log-likelihood ratio (i.e. G-test)
terms = 2.0 * special.xlogy(f_obs, f_obs / f_exp)
elif lambda_ == -1:
# Modified log-likelihood ratio
terms = 2.0 * special.xlogy(f_exp, f_exp / f_obs)
else:
# General Cressie-Read power divergence.
terms = f_obs * ((f_obs / f_exp)**lambda_ - 1)
terms /= 0.5 * lambda_ * (lambda_ + 1)
stat = terms.sum(axis=axis)
num_obs = _count(terms, axis=axis)
ddof = asarray(ddof)
p = distributions.chi2.sf(stat, num_obs - 1 - ddof)
return Power_divergenceResult(stat, p)
def chisquare(f_obs, f_exp=None, ddof=0, axis=0):
"""
Calculate a one-way chi square test.
The chi square test tests the null hypothesis that the categorical data
has the given frequencies.
Parameters
----------
f_obs : array_like
Observed frequencies in each category.
f_exp : array_like, optional
Expected frequencies in each category. By default the categories are
assumed to be equally likely.
ddof : int, optional
"Delta degrees of freedom": adjustment to the degrees of freedom
for the p-value. The p-value is computed using a chi-squared
distribution with ``k - 1 - ddof`` degrees of freedom, where `k`
is the number of observed frequencies. The default value of `ddof`
is 0.
axis : int or None, optional
The axis of the broadcast result of `f_obs` and `f_exp` along which to
apply the test. If axis is None, all values in `f_obs` are treated
as a single data set. Default is 0.
Returns
-------
chisq : float or ndarray
The chi-squared test statistic. The value is a float if `axis` is
None or `f_obs` and `f_exp` are 1-D.
p : float or ndarray
The p-value of the test. The value is a float if `ddof` and the
return value `chisq` are scalars.
See Also
--------
power_divergence
mstats.chisquare
Notes
-----
This test is invalid when the observed or expected frequencies in each
category are too small. A typical rule is that all of the observed
and expected frequencies should be at least 5.
The default degrees of freedom, k-1, are for the case when no parameters
of the distribution are estimated. If p parameters are estimated by
efficient maximum likelihood then the correct degrees of freedom are
k-1-p. If the parameters are estimated in a different way, then the
dof can be between k-1-p and k-1. However, it is also possible that
the asymptotic distribution is not a chisquare, in which case this
test is not appropriate.
References
----------
.. [1] Lowry, Richard. "Concepts and Applications of Inferential
Statistics". Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html
.. [2] "Chi-squared test", http://en.wikipedia.org/wiki/Chi-squared_test
Examples
--------
When just `f_obs` is given, it is assumed that the expected frequencies
are uniform and given by the mean of the observed frequencies.
>>> from scipy.stats import chisquare
>>> chisquare([16, 18, 16, 14, 12, 12])
(2.0, 0.84914503608460956)
With `f_exp` the expected frequencies can be given.
>>> chisquare([16, 18, 16, 14, 12, 12], f_exp=[16, 16, 16, 16, 16, 8])
(3.5, 0.62338762774958223)
When `f_obs` is 2-D, by default the test is applied to each column.
>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
>>> obs.shape
(6, 2)
>>> chisquare(obs)
(array([ 2. , 6.66666667]), array([ 0.84914504, 0.24663415]))
By setting ``axis=None``, the test is applied to all data in the array,
which is equivalent to applying the test to the flattened array.
>>> chisquare(obs, axis=None)
(23.31034482758621, 0.015975692534127565)
>>> chisquare(obs.ravel())
(23.31034482758621, 0.015975692534127565)
`ddof` is the change to make to the default degrees of freedom.
>>> chisquare([16, 18, 16, 14, 12, 12], ddof=1)
(2.0, 0.73575888234288467)
The calculation of the p-values is done by broadcasting the
chi-squared statistic with `ddof`.
>>> chisquare([16, 18, 16, 14, 12, 12], ddof=[0,1,2])
(2.0, array([ 0.84914504, 0.73575888, 0.5724067 ]))
`f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has
shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting
`f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared
statistics, we use ``axis=1``:
>>> chisquare([16, 18, 16, 14, 12, 12],
... f_exp=[[16, 16, 16, 16, 16, 8], [8, 20, 20, 16, 12, 12]],
... axis=1)
(array([ 3.5 , 9.25]), array([ 0.62338763, 0.09949846]))
"""
return power_divergence(f_obs, f_exp=f_exp, ddof=ddof, axis=axis,
lambda_="pearson")
Ks_2sampResult = namedtuple('Ks_2sampResult', ('statistic', 'pvalue'))
def ks_2samp(data1, data2):
"""
Compute the Kolmogorov-Smirnov statistic on 2 samples.
This is a two-sided test for the null hypothesis that 2 independent samples
are drawn from the same continuous distribution.
Parameters
----------
data1, data2 : sequence of 1-D ndarrays
two arrays of sample observations assumed to be drawn from a continuous
distribution, sample sizes can be different
Returns
-------
statistic : float
KS statistic
pvalue : float
two-tailed p-value
Notes
-----
This tests whether 2 samples are drawn from the same distribution. Note
that, like in the case of the one-sample K-S test, the distribution is
assumed to be continuous.
This is the two-sided test, one-sided tests are not implemented.
The test uses the two-sided asymptotic Kolmogorov-Smirnov distribution.
If the K-S statistic is small or the p-value is high, then we cannot
reject the hypothesis that the distributions of the two samples
are the same.
Examples
--------
>>> from scipy import stats
>>> np.random.seed(12345678) #fix random seed to get the same result
>>> n1 = 200 # size of first sample
>>> n2 = 300 # size of second sample
For a different distribution, we can reject the null hypothesis since the
pvalue is below 1%:
>>> rvs1 = stats.norm.rvs(size=n1, loc=0., scale=1)
>>> rvs2 = stats.norm.rvs(size=n2, loc=0.5, scale=1.5)
>>> stats.ks_2samp(rvs1, rvs2)
(0.20833333333333337, 4.6674975515806989e-005)
For a slightly different distribution, we cannot reject the null hypothesis
at a 10% or lower alpha since the p-value at 0.144 is higher than 10%
>>> rvs3 = stats.norm.rvs(size=n2, loc=0.01, scale=1.0)
>>> stats.ks_2samp(rvs1, rvs3)
(0.10333333333333333, 0.14498781825751686)
For an identical distribution, we cannot reject the null hypothesis since
the p-value is high, 41%:
>>> rvs4 = stats.norm.rvs(size=n2, loc=0.0, scale=1.0)
>>> stats.ks_2samp(rvs1, rvs4)
(0.07999999999999996, 0.41126949729859719)
"""
data1 = np.sort(data1)
data2 = np.sort(data2)
n1 = data1.shape[0]
n2 = data2.shape[0]
data_all = np.concatenate([data1, data2])
cdf1 = np.searchsorted(data1, data_all, side='right') / (1.0*n1)
cdf2 = np.searchsorted(data2, data_all, side='right') / (1.0*n2)
d = np.max(np.absolute(cdf1 - cdf2))
# Note: d absolute not signed distance
en = np.sqrt(n1 * n2 / float(n1 + n2))
try:
prob = distributions.kstwobign.sf((en + 0.12 + 0.11 / en) * d)
except:
prob = 1.0
return Ks_2sampResult(d, prob)
def tiecorrect(rankvals):
"""
Tie correction factor for ties in the Mann-Whitney U and
Kruskal-Wallis H tests.
Parameters
----------
rankvals : array_like
A 1-D sequence of ranks. Typically this will be the array
returned by `stats.rankdata`.
Returns
-------
factor : float
Correction factor for U or H.
See Also
--------
rankdata : Assign ranks to the data
mannwhitneyu : Mann-Whitney rank test
kruskal : Kruskal-Wallis H test
References
----------
.. [1] Siegel, S. (1956) Nonparametric Statistics for the Behavioral
Sciences. New York: McGraw-Hill.
Examples
--------
>>> from scipy.stats import tiecorrect, rankdata
>>> tiecorrect([1, 2.5, 2.5, 4])
0.9
>>> ranks = rankdata([1, 3, 2, 4, 5, 7, 2, 8, 4])
>>> ranks
array([ 1. , 4. , 2.5, 5.5, 7. , 8. , 2.5, 9. , 5.5])
>>> tiecorrect(ranks)
0.9833333333333333
"""
arr = np.sort(rankvals)
idx = np.nonzero(np.r_[True, arr[1:] != arr[:-1], True])[0]
cnt = np.diff(idx).astype(np.float64)
size = np.float64(arr.size)
return 1.0 if size < 2 else 1.0 - (cnt**3 - cnt).sum() / (size**3 - size)
MannwhitneyuResult = namedtuple('MannwhitneyuResult', ('statistic', 'pvalue'))
def mannwhitneyu(x, y, use_continuity=True, alternative=None):
"""
Compute the Mann-Whitney rank test on samples x and y.
Parameters
----------
x, y : array_like
Array of samples, should be one-dimensional.
use_continuity : bool, optional
Whether a continuity correction (1/2.) should be taken into
account. Default is True.
alternative : None (deprecated), 'less', 'two-sided', or 'greater'
Whether to get the p-value for the one-sided hypothesis ('less'
or 'greater') or for the two-sided hypothesis ('two-sided').
Defaults to None, which results in a p-value half the size of
the 'two-sided' p-value and a different U statistic. The
default behavior is not the same as using 'less' or 'greater':
it only exists for backward compatibility and is deprecated.
Returns
-------
statistic : float
The Mann-Whitney U statistic, equal to min(U for x, U for y) if
`alternative` is equal to None (deprecated; exists for backward
compatibility), and U for y otherwise.
pvalue : float
p-value assuming an asymptotic normal distribution. One-sided or
two-sided, depending on the choice of `alternative`.
Notes
-----
Use only when the number of observation in each sample is > 20 and
you have 2 independent samples of ranks. Mann-Whitney U is
significant if the u-obtained is LESS THAN or equal to the critical
value of U.
This test corrects for ties and by default uses a continuity correction.
References
----------
.. [1] https://en.wikipedia.org/wiki/Mann-Whitney_U_test
.. [2] H.B. Mann and D.R. Whitney, "On a Test of Whether one of Two Random
Variables is Stochastically Larger than the Other," The Annals of
Mathematical Statistics, vol. 18, no. 1, pp. 50-60, 1947.
"""
if alternative is None:
warnings.warn("Calling `mannwhitneyu` without specifying "
"`alternative` is deprecated.", DeprecationWarning)
x = np.asarray(x)
y = np.asarray(y)
n1 = len(x)
n2 = len(y)
ranked = rankdata(np.concatenate((x, y)))
rankx = ranked[0:n1] # get the x-ranks
u1 = n1*n2 + (n1*(n1+1))/2.0 - np.sum(rankx, axis=0) # calc U for x
u2 = n1*n2 - u1 # remainder is U for y
T = tiecorrect(ranked)
if T == 0:
raise ValueError('All numbers are identical in mannwhitneyu')
sd = np.sqrt(T * n1 * n2 * (n1+n2+1) / 12.0)
meanrank = n1*n2/2.0 + 0.5 * use_continuity
if alternative is None or alternative == 'two-sided':
bigu = max(u1, u2)
elif alternative == 'less':
bigu = u1
elif alternative == 'greater':
bigu = u2
else:
raise ValueError("alternative should be None, 'less', 'greater' "
"or 'two-sided'")
z = (bigu - meanrank) / sd
if alternative is None:
# This behavior, equal to half the size of the two-sided
# p-value, is deprecated.
p = distributions.norm.sf(abs(z))
elif alternative == 'two-sided':
p = 2 * distributions.norm.sf(abs(z))
else:
p = distributions.norm.sf(z)
u = u2
# This behavior is deprecated.
if alternative is None:
u = min(u1, u2)
return MannwhitneyuResult(u, p)
RanksumsResult = namedtuple('RanksumsResult', ('statistic', 'pvalue'))
def ranksums(x, y):
"""
Compute the Wilcoxon rank-sum statistic for two samples.
The Wilcoxon rank-sum test tests the null hypothesis that two sets
of measurements are drawn from the same distribution. The alternative
hypothesis is that values in one sample are more likely to be
larger than the values in the other sample.
This test should be used to compare two samples from continuous
distributions. It does not handle ties between measurements
in x and y. For tie-handling and an optional continuity correction
see `scipy.stats.mannwhitneyu`.
Parameters
----------
x,y : array_like
The data from the two samples
Returns
-------
statistic : float
The test statistic under the large-sample approximation that the
rank sum statistic is normally distributed
pvalue : float
The two-sided p-value of the test
References
----------
.. [1] http://en.wikipedia.org/wiki/Wilcoxon_rank-sum_test
"""
x, y = map(np.asarray, (x, y))
n1 = len(x)
n2 = len(y)
alldata = np.concatenate((x, y))
ranked = rankdata(alldata)
x = ranked[:n1]
s = np.sum(x, axis=0)
expected = n1 * (n1+n2+1) / 2.0
z = (s - expected) / np.sqrt(n1*n2*(n1+n2+1)/12.0)
prob = 2 * distributions.norm.sf(abs(z))
return RanksumsResult(z, prob)
KruskalResult = namedtuple('KruskalResult', ('statistic', 'pvalue'))
def kruskal(*args, **kwargs):
"""
Compute the Kruskal-Wallis H-test for independent samples
The Kruskal-Wallis H-test tests the null hypothesis that the population
median of all of the groups are equal. It is a non-parametric version of
ANOVA. The test works on 2 or more independent samples, which may have
different sizes. Note that rejecting the null hypothesis does not
indicate which of the groups differs. Post-hoc comparisons between
groups are required to determine which groups are different.
Parameters
----------
sample1, sample2, ... : array_like
Two or more arrays with the sample measurements can be given as
arguments.
nan_policy : {'propagate', 'raise', 'omit'}, optional
Defines how to handle when input contains nan. 'propagate' returns nan,
'raise' throws an error, 'omit' performs the calculations ignoring nan
values. Default is 'propagate'.
Returns
-------
statistic : float
The Kruskal-Wallis H statistic, corrected for ties
pvalue : float
The p-value for the test using the assumption that H has a chi
square distribution
See Also
--------
f_oneway : 1-way ANOVA
mannwhitneyu : Mann-Whitney rank test on two samples.
friedmanchisquare : Friedman test for repeated measurements
Notes
-----
Due to the assumption that H has a chi square distribution, the number
of samples in each group must not be too small. A typical rule is
that each sample must have at least 5 measurements.
References
----------
.. [1] W. H. Kruskal & W. W. Wallis, "Use of Ranks in
One-Criterion Variance Analysis", Journal of the American Statistical
Association, Vol. 47, Issue 260, pp. 583-621, 1952.
.. [2] http://en.wikipedia.org/wiki/Kruskal-Wallis_one-way_analysis_of_variance
Examples
--------
>>> from scipy import stats
>>> x = [1, 3, 5, 7, 9]
>>> y = [2, 4, 6, 8, 10]
>>> stats.kruskal(x, y)
KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895)
>>> x = [1, 1, 1]
>>> y = [2, 2, 2]
>>> z = [2, 2]
>>> stats.kruskal(x, y, z)
KruskalResult(statistic=7.0, pvalue=0.0301973834223185)
"""
args = list(map(np.asarray, args))
num_groups = len(args)
if num_groups < 2:
raise ValueError("Need at least two groups in stats.kruskal()")
for arg in args:
if arg.size == 0:
return KruskalResult(np.nan, np.nan)
n = np.asarray(list(map(len, args)))
if 'nan_policy' in kwargs.keys():
if kwargs['nan_policy'] not in ('propagate', 'raise', 'omit'):
raise ValueError("nan_policy must be 'propagate', "
"'raise' or'omit'")
else:
nan_policy = kwargs['nan_policy']
else:
nan_policy = 'propagate'
contains_nan = False
for arg in args:
cn = _contains_nan(arg, nan_policy)
if cn[0]:
contains_nan = True
break
if contains_nan and nan_policy == 'omit':
for a in args:
a = ma.masked_invalid(a)
return mstats_basic.kruskal(*args)
if contains_nan and nan_policy == 'propagate':
return KruskalResult(np.nan, np.nan)
alldata = np.concatenate(args)
ranked = rankdata(alldata)
ties = tiecorrect(ranked)
if ties == 0:
raise ValueError('All numbers are identical in kruskal')
# Compute sum^2/n for each group and sum
j = np.insert(np.cumsum(n), 0, 0)
ssbn = 0
for i in range(num_groups):
ssbn += _square_of_sums(ranked[j[i]:j[i+1]]) / float(n[i])
totaln = np.sum(n)
h = 12.0 / (totaln * (totaln + 1)) * ssbn - 3 * (totaln + 1)
df = num_groups - 1
h /= ties
return KruskalResult(h, distributions.chi2.sf(h, df))
FriedmanchisquareResult = namedtuple('FriedmanchisquareResult',
('statistic', 'pvalue'))
def friedmanchisquare(*args):
"""
Compute the Friedman test for repeated measurements
The Friedman test tests the null hypothesis that repeated measurements of
the same individuals have the same distribution. It is often used
to test for consistency among measurements obtained in different ways.
For example, if two measurement techniques are used on the same set of
individuals, the Friedman test can be used to determine if the two
measurement techniques are consistent.
Parameters
----------
measurements1, measurements2, measurements3... : array_like
Arrays of measurements. All of the arrays must have the same number
of elements. At least 3 sets of measurements must be given.
Returns
-------
statistic : float
the test statistic, correcting for ties
pvalue : float
the associated p-value assuming that the test statistic has a chi
squared distribution
Notes
-----
Due to the assumption that the test statistic has a chi squared
distribution, the p-value is only reliable for n > 10 and more than
6 repeated measurements.
References
----------
.. [1] http://en.wikipedia.org/wiki/Friedman_test
"""
k = len(args)
if k < 3:
raise ValueError('Less than 3 levels. Friedman test not appropriate.')
n = len(args[0])
for i in range(1, k):
if len(args[i]) != n:
raise ValueError('Unequal N in friedmanchisquare. Aborting.')
# Rank data
data = np.vstack(args).T
data = data.astype(float)
for i in range(len(data)):
data[i] = rankdata(data[i])
# Handle ties
ties = 0
for i in range(len(data)):
replist, repnum = find_repeats(array(data[i]))
for t in repnum:
ties += t * (t*t - 1)
c = 1 - ties / float(k*(k*k - 1)*n)
ssbn = np.sum(data.sum(axis=0)**2)
chisq = (12.0 / (k*n*(k+1)) * ssbn - 3*n*(k+1)) / c
return FriedmanchisquareResult(chisq, distributions.chi2.sf(chisq, k - 1))
def combine_pvalues(pvalues, method='fisher', weights=None):
"""
Methods for combining the p-values of independent tests bearing upon the
same hypothesis.
Parameters
----------
pvalues : array_like, 1-D
Array of p-values assumed to come from independent tests.
method : {'fisher', 'stouffer'}, optional
Name of method to use to combine p-values. The following methods are
available:
- "fisher": Fisher's method (Fisher's combined probability test),
the default.
- "stouffer": Stouffer's Z-score method.
weights : array_like, 1-D, optional
Optional array of weights used only for Stouffer's Z-score method.
Returns
-------
statistic: float
The statistic calculated by the specified method:
- "fisher": The chi-squared statistic
- "stouffer": The Z-score
pval: float
The combined p-value.
Notes
-----
Fisher's method (also known as Fisher's combined probability test) [1]_ uses
a chi-squared statistic to compute a combined p-value. The closely related
Stouffer's Z-score method [2]_ uses Z-scores rather than p-values. The
advantage of Stouffer's method is that it is straightforward to introduce
weights, which can make Stouffer's method more powerful than Fisher's
method when the p-values are from studies of different size [3]_ [4]_.
Fisher's method may be extended to combine p-values from dependent tests
[5]_. Extensions such as Brown's method and Kost's method are not currently
implemented.
.. versionadded:: 0.15.0
References
----------
.. [1] https://en.wikipedia.org/wiki/Fisher%27s_method
.. [2] http://en.wikipedia.org/wiki/Fisher's_method#Relation_to_Stouffer.27s_Z-score_method
.. [3] Whitlock, M. C. "Combining probability from independent tests: the
weighted Z-method is superior to Fisher's approach." Journal of
Evolutionary Biology 18, no. 5 (2005): 1368-1373.
.. [4] Zaykin, Dmitri V. "Optimally weighted Z-test is a powerful method
for combining probabilities in meta-analysis." Journal of
Evolutionary Biology 24, no. 8 (2011): 1836-1841.
.. [5] https://en.wikipedia.org/wiki/Extensions_of_Fisher%27s_method
"""
pvalues = np.asarray(pvalues)
if pvalues.ndim != 1:
raise ValueError("pvalues is not 1-D")
if method == 'fisher':
Xsq = -2 * np.sum(np.log(pvalues))
pval = distributions.chi2.sf(Xsq, 2 * len(pvalues))
return (Xsq, pval)
elif method == 'stouffer':
if weights is None:
weights = np.ones_like(pvalues)
elif len(weights) != len(pvalues):
raise ValueError("pvalues and weights must be of the same size.")
weights = np.asarray(weights)
if weights.ndim != 1:
raise ValueError("weights is not 1-D")
Zi = distributions.norm.isf(pvalues)
Z = np.dot(weights, Zi) / np.linalg.norm(weights)
pval = distributions.norm.sf(Z)
return (Z, pval)
else:
raise ValueError(
"Invalid method '%s'. Options are 'fisher' or 'stouffer'", method)
#####################################
# PROBABILITY CALCULATIONS #
#####################################
def _betai(a, b, x):
x = np.asarray(x)
x = np.where(x < 1.0, x, 1.0) # if x > 1 then return 1.0
return special.betainc(a, b, x)
#####################################
# STATISTICAL DISTANCES #
#####################################
def wasserstein_distance(u_values, v_values, u_weights=None, v_weights=None):
r"""
Compute the first Wasserstein distance between two 1D distributions.
This distance is also known as the earth mover's distance, since it can be
seen as the minimum amount of "work" required to transform :math:`u` into
:math:`v`, where "work" is measured as the amount of distribution weight
that must be moved, multiplied by the distance it has to be moved.
.. versionadded:: 1.0.0
Parameters
----------
u_values, v_values : array_like
Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional
Weight for each value. If unspecified, each value is assigned the same
weight.
`u_weights` (resp. `v_weights`) must have the same length as
`u_values` (resp. `v_values`). If the weight sum differs from 1, it
must still be positive and finite so that the weights can be normalized
to sum to 1.
Returns
-------
distance : float
The computed distance between the distributions.
Notes
-----
The first Wasserstein distance between the distributions :math:`u` and
:math:`v` is:
.. math::
l_1 (u, v) = \inf_{\pi \in \Gamma (u, v)} \int_{\mathbb{R} \times
\mathbb{R}} |x-y| \mathrm{d} \pi (x, y)
where :math:`\Gamma (u, v)` is the set of (probability) distributions on
:math:`\mathbb{R} \times \mathbb{R}` whose marginals are :math:`u` and
:math:`v` on the first and second factors respectively.
If :math:`U` and :math:`V` are the respective CDFs of :math:`u` and
:math:`v`, this distance also equals to:
.. math::
l_1(u, v) = \int_{-\infty}^{+\infty} |U-V|
See [2]_ for a proof of the equivalence of both definitions.
The input distributions can be empirical, therefore coming from samples
whose values are effectively inputs of the function, or they can be seen as
generalized functions, in which case they are weighted sums of Dirac delta
functions located at the specified values.
References
----------
.. [1] "Wasserstein metric", http://en.wikipedia.org/wiki/Wasserstein_metric
.. [2] Ramdas, Garcia, Cuturi "On Wasserstein Two Sample Testing and Related
Families of Nonparametric Tests" (2015). :arXiv:`1509.02237`.
Examples
--------
>>> from scipy.stats import wasserstein_distance
>>> wasserstein_distance([0, 1, 3], [5, 6, 8])
5.0
>>> wasserstein_distance([0, 1], [0, 1], [3, 1], [2, 2])
0.25
>>> wasserstein_distance([3.4, 3.9, 7.5, 7.8], [4.5, 1.4],
... [1.4, 0.9, 3.1, 7.2], [3.2, 3.5])
4.0781331438047861
"""
return _cdf_distance(1, u_values, v_values, u_weights, v_weights)
def energy_distance(u_values, v_values, u_weights=None, v_weights=None):
r"""
Compute the energy distance between two 1D distributions.
.. versionadded:: 1.0.0
Parameters
----------
u_values, v_values : array_like
Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional
Weight for each value. If unspecified, each value is assigned the same
weight.
`u_weights` (resp. `v_weights`) must have the same length as
`u_values` (resp. `v_values`). If the weight sum differs from 1, it
must still be positive and finite so that the weights can be normalized
to sum to 1.
Returns
-------
distance : float
The computed distance between the distributions.
Notes
-----
The energy distance between two distributions :math:`u` and :math:`v`, whose
respective CDFs are :math:`U` and :math:`V`, equals to:
.. math::
D(u, v) = \left( 2\mathbb E|X - Y| - \mathbb E|X - X'| -
\mathbb E|Y - Y'| \right)^{1/2}
where :math:`X` and :math:`X'` (resp. :math:`Y` and :math:`Y'`) are
independent random variables whose probability distribution is :math:`u`
(resp. :math:`v`).
As shown in [2]_, for one-dimensional real-valued variables, the energy
distance is linked to the non-distribution-free version of the Cramer-von
Mises distance:
.. math::
D(u, v) = \sqrt{2} l_2(u, v) = \left( 2 \int_{-\infty}^{+\infty} (U-V)^2
\right)^{1/2}
Note that the common Cramer-von Mises criterion uses the distribution-free
version of the distance. See [2]_ (section 2), for more details about both
versions of the distance.
The input distributions can be empirical, therefore coming from samples
whose values are effectively inputs of the function, or they can be seen as
generalized functions, in which case they are weighted sums of Dirac delta
functions located at the specified values.
References
----------
.. [1] "Energy distance", https://en.wikipedia.org/wiki/Energy_distance
.. [2] Szekely "E-statistics: The energy of statistical samples." Bowling
Green State University, Department of Mathematics and Statistics,
Technical Report 02-16 (2002).
.. [3] Rizzo, Szekely "Energy distance." Wiley Interdisciplinary Reviews:
Computational Statistics, 8(1):27-38 (2015).
.. [4] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer,
Munos "The Cramer Distance as a Solution to Biased Wasserstein
Gradients" (2017). :arXiv:`1705.10743`.
Examples
--------
>>> from scipy.stats import energy_distance
>>> energy_distance([0], [2])
2.0000000000000004
>>> energy_distance([0, 8], [0, 8], [3, 1], [2, 2])
1.0000000000000002
>>> energy_distance([0.7, 7.4, 2.4, 6.8], [1.4, 8. ],
... [2.1, 4.2, 7.4, 8. ], [7.6, 8.8])
0.88003340976158217
"""
return np.sqrt(2) * _cdf_distance(2, u_values, v_values,
u_weights, v_weights)
def _cdf_distance(p, u_values, v_values, u_weights=None, v_weights=None):
r"""
Compute, between two one-dimensional distributions :math:`u` and
:math:`v`, whose respective CDFs are :math:`U` and :math:`V`, the
statistical distance that is defined as:
.. math::
l_p(u, v) = \left( \int_{-\infty}^{+\infty} |U-V|^p \right)^{1/p}
p is a positive parameter; p = 1 gives the Wasserstein distance, p = 2
gives the energy distance.
Parameters
----------
u_values, v_values : array_like
Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional
Weight for each value. If unspecified, each value is assigned the same
weight.
`u_weights` (resp. `v_weights`) must have the same length as
`u_values` (resp. `v_values`). If the weight sum differs from 1, it
must still be positive and finite so that the weights can be normalized
to sum to 1.
Returns
-------
distance : float
The computed distance between the distributions.
Notes
-----
The input distributions can be empirical, therefore coming from samples
whose values are effectively inputs of the function, or they can be seen as
generalized functions, in which case they are weighted sums of Dirac delta
functions located at the specified values.
References
----------
.. [1] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer,
Munos "The Cramer Distance as a Solution to Biased Wasserstein
Gradients" (2017). :arXiv:`1705.10743`.
"""
u_values, u_weights = _validate_distribution(u_values, u_weights)
v_values, v_weights = _validate_distribution(v_values, v_weights)
u_sorter = np.argsort(u_values)
v_sorter = np.argsort(v_values)
all_values = np.concatenate((u_values, v_values))
all_values.sort(kind='mergesort')
# Compute the differences between pairs of successive values of u and v.
deltas = np.diff(all_values)
# Get the respective positions of the values of u and v among the values of
# both distributions.
u_cdf_indices = u_values[u_sorter].searchsorted(all_values[:-1], 'right')
v_cdf_indices = v_values[v_sorter].searchsorted(all_values[:-1], 'right')
# Calculate the CDFs of u and v using their weights, if specified.
if u_weights is None:
u_cdf = u_cdf_indices / u_values.size
else:
u_sorted_cumweights = np.concatenate(([0],
np.cumsum(u_weights[u_sorter])))
u_cdf = u_sorted_cumweights[u_cdf_indices] / u_sorted_cumweights[-1]
if v_weights is None:
v_cdf = v_cdf_indices / v_values.size
else:
v_sorted_cumweights = np.concatenate(([0],
np.cumsum(v_weights[v_sorter])))
v_cdf = v_sorted_cumweights[v_cdf_indices] / v_sorted_cumweights[-1]
# Compute the value of the integral based on the CDFs.
# If p = 1 or p = 2, we avoid using np.power, which introduces an overhead
# of about 15%.
if p == 1:
return np.sum(np.multiply(np.abs(u_cdf - v_cdf), deltas))
if p == 2:
return np.sqrt(np.sum(np.multiply(np.square(u_cdf - v_cdf), deltas)))
return np.power(np.sum(np.multiply(np.power(np.abs(u_cdf - v_cdf), p),
deltas)), 1/p)
def _validate_distribution(values, weights):
"""
Validate the values and weights from a distribution input of `cdf_distance`
and return them as ndarray objects.
Parameters
----------
values : array_like
Values observed in the (empirical) distribution.
weights : array_like
Weight for each value.
Returns
-------
values : ndarray
Values as ndarray.
weights : ndarray
Weights as ndarray.
"""
# Validate the value array.
values = np.asarray(values, dtype=float)
if len(values) == 0:
raise ValueError("Distribution can't be empty.")
# Validate the weight array, if specified.
if weights is not None:
weights = np.asarray(weights, dtype=float)
if len(weights) != len(values):
raise ValueError('Value and weight array-likes for the same '
'empirical distribution must be of the same size.')
if np.any(weights < 0):
raise ValueError('All weights must be non-negative.')
if not 0 < np.sum(weights) < np.inf:
raise ValueError('Weight array-like sum must be positive and '
'finite. Set as None for an equal distribution of '
'weight.')
return values, weights
return values, None
#####################################
# SUPPORT FUNCTIONS #
#####################################
RepeatedResults = namedtuple('RepeatedResults', ('values', 'counts'))
def find_repeats(arr):
"""
Find repeats and repeat counts.
Parameters
----------
arr : array_like
Input array. This is cast to float64.
Returns
-------
values : ndarray
The unique values from the (flattened) input that are repeated.
counts : ndarray
Number of times the corresponding 'value' is repeated.
Notes
-----
In numpy >= 1.9 `numpy.unique` provides similar functionality. The main
difference is that `find_repeats` only returns repeated values.
Examples
--------
>>> from scipy import stats
>>> stats.find_repeats([2, 1, 2, 3, 2, 2, 5])
RepeatedResults(values=array([2.]), counts=array([4]))
>>> stats.find_repeats([[10, 20, 1, 2], [5, 5, 4, 4]])
RepeatedResults(values=array([4., 5.]), counts=array([2, 2]))
"""
# Note: always copies.
return RepeatedResults(*_find_repeats(np.array(arr, dtype=np.float64)))
def _sum_of_squares(a, axis=0):
"""
Square each element of the input array, and return the sum(s) of that.
Parameters
----------
a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate. Default is 0. If None, compute over
the whole array `a`.
Returns
-------
sum_of_squares : ndarray
The sum along the given axis for (a**2).
See also
--------
_square_of_sums : The square(s) of the sum(s) (the opposite of
`_sum_of_squares`).
"""
a, axis = _chk_asarray(a, axis)
return np.sum(a*a, axis)
def _square_of_sums(a, axis=0):
"""
Sum elements of the input array, and return the square(s) of that sum.
Parameters
----------
a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate. Default is 0. If None, compute over
the whole array `a`.
Returns
-------
square_of_sums : float or ndarray
The square of the sum over `axis`.
See also
--------
_sum_of_squares : The sum of squares (the opposite of `square_of_sums`).
"""
a, axis = _chk_asarray(a, axis)
s = np.sum(a, axis)
if not np.isscalar(s):
return s.astype(float) * s
else:
return float(s) * s
def rankdata(a, method='average'):
"""
Assign ranks to data, dealing with ties appropriately.
Ranks begin at 1. The `method` argument controls how ranks are assigned
to equal values. See [1]_ for further discussion of ranking methods.
Parameters
----------
a : array_like
The array of values to be ranked. The array is first flattened.
method : str, optional
The method used to assign ranks to tied elements.
The options are 'average', 'min', 'max', 'dense' and 'ordinal'.
'average':
The average of the ranks that would have been assigned to
all the tied values is assigned to each value.
'min':
The minimum of the ranks that would have been assigned to all
the tied values is assigned to each value. (This is also
referred to as "competition" ranking.)
'max':
The maximum of the ranks that would have been assigned to all
the tied values is assigned to each value.
'dense':
Like 'min', but the rank of the next highest element is assigned
the rank immediately after those assigned to the tied elements.
'ordinal':
All values are given a distinct rank, corresponding to the order
that the values occur in `a`.
The default is 'average'.
Returns
-------
ranks : ndarray
An array of length equal to the size of `a`, containing rank
scores.
References
----------
.. [1] "Ranking", http://en.wikipedia.org/wiki/Ranking
Examples
--------
>>> from scipy.stats import rankdata
>>> rankdata([0, 2, 3, 2])
array([ 1. , 2.5, 4. , 2.5])
>>> rankdata([0, 2, 3, 2], method='min')
array([ 1, 2, 4, 2])
>>> rankdata([0, 2, 3, 2], method='max')
array([ 1, 3, 4, 3])
>>> rankdata([0, 2, 3, 2], method='dense')
array([ 1, 2, 3, 2])
>>> rankdata([0, 2, 3, 2], method='ordinal')
array([ 1, 2, 4, 3])
"""
if method not in ('average', 'min', 'max', 'dense', 'ordinal'):
raise ValueError('unknown method "{0}"'.format(method))
arr = np.ravel(np.asarray(a))
algo = 'mergesort' if method == 'ordinal' else 'quicksort'
sorter = np.argsort(arr, kind=algo)
inv = np.empty(sorter.size, dtype=np.intp)
inv[sorter] = np.arange(sorter.size, dtype=np.intp)
if method == 'ordinal':
return inv + 1
arr = arr[sorter]
obs = np.r_[True, arr[1:] != arr[:-1]]
dense = obs.cumsum()[inv]
if method == 'dense':
return dense
# cumulative counts of each unique value
count = np.r_[np.nonzero(obs)[0], len(obs)]
if method == 'max':
return count[dense]
if method == 'min':
return count[dense - 1] + 1
# average method
return .5 * (count[dense] + count[dense - 1] + 1)
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